Is this type of behavior a type of rental scam?

Need some guidance here. Usually pretty good at spotting scams but this one's got me confused as it could very well just be a very 'cautious' property owner trying to rent a place, but the 'leave the money order blank' and no mention of a rental agreement and lack of contact information are making me very suspicious.

As discussed in comments, it would be difficult to say conclusively that it is a scam or the owner is afraid of getting scammed and hence being cautious.

The best in such situations is to walk away from the deal.

In this case you can check further to determine if this is a scam or not by;

• Asking him for identification. There shouldn't be any issue showing the identification.
• Asking him to show ownership of the place, be it utility bills or Lease agreement with the owner and permission to sub-lease
• Ask him what other mode of payments will be accept other than Money Order [or cash]
• Ask for a draft lease document so that you can read through and agree to give the deposit and sign the lease
• Ask him for the address of the house / apartment again as you would like to visit it again before you sign the deal

If it is a scam should I Change my bank account number, which I included on the application?

Preferably yes. If it is a hassle, leave very less funds and move everything to new account.

Should I get a police report?

You can try. Generally since it cannot be conclusively be established that it is fraud and the amounts involved, police may not do anything much.

You ask this:

Is this some sort of very elaborate scam?

The answer is simple:

No, this is not elaborate. But yes, this is utterly a scam.

Without knowing any deep details you simply need to understand this basic rule of life when it comes to stuff like this:

If a situation feels unconformable and “scammy” to you, then it is something you need to avoid.

That is really the beginning, middle and end of this. A scammer needs a willing participant and if you subjugate yourself to their needs despite every “this feels wrong” instinct going off in your head, then you are helping the scammer scam you rather than helping yourself.

Reading what you have posted here are the red flags that show you know this without needing to ask anyone:

• “Instead, he asked me to meet him at a different, public place near by, and then had me follow him back to the house.” This is almost understandable. Many people with concerns about public safety will ask to meet someone at a neutral location. That said, someone attempting to engage in a real estate transaction who is hiding details of the location should immediately be suspect.
• “The person had me fill out an 'application' that looked like a generic apartment application that had been retrieved from the web.” If you feel the application is generic and it asks for tons of details, you should push back on it. The reality is in many legitimate real estate transactions details like this are par for the course. But honestly I have never heard of anything like this for a sub-lease where people are sharing the apartment. I would have simply stated, “You are asking for too many details for now. I like this place. What can we do to make this happen without these details.” In many legitimate transactions, person on the other side of this transaction will accommodate the request.
• “At this point, I don't even know the guy's last name. Just the first name.” Massive red flag. If I told you I gave personal information to someone who I barely knew who only provided me with their first name you would tell me I am crazy, right?
• “I tried running a reverse phone lookup on the number, and it says it's a VOIP phone with no name listed.” Another red flag. While many people use VOIP numbers as legitimate numbers, many use them as “burner” numbers that are easily untraceable and ultimately disposable. Many people engaged in real estate might have two numbers—one personal and one work—but they are often legitimate numbers that can easily be traced. A VOIP number on the top of the rest of this?
• “He asked me to pay him the deposit in a money order, and to leave the money order fields blank.” Paying via a money order is not suspect. But leaving the fields blank? Really questionable. Why would someone whose last name you don’t know want a money order with the payee area blank?

And so on and so on.

But it still comes back to the statement I made at the outset: “If a situation feels unconformable and “scammy” to you, then it is something you need to avoid.”

Even if somehow this were all 100% legitimate, would you feel comfortable living in a place and paying rent to someone who behaves in such a manner? Would you actually be able to sleep each night feeling secure in such a situation? Remember, it might be better to pay $700-900 a month instead of $500-600 to simply have basic peace of mind.

Money is money. But peace of mind and comfort for your own well being are priceless. Always put yourself first and don’t ignore your own instincts.

Hmm, I can think of legitimate reasons for all of this.

Meet at a public place: Maybe he's just being cautious. If he's living in this house and is renting out a room, maybe he doesn't want to give out his address to any random person who contacts him. He figures he wants to at least meet the person first.

Generic application downloaded from Internet: I don't find that suspicious at all. If this is the only place he's putting up for rent, and especially if he's never rented out a room before, he likely has no idea what to ask on an application. So he downloads something from the Internet.

Smartmove: So maybe he had you fill out this application, and then when he tried to run a background check, he found there was an easier way. Again, I don't find that suspicious at all.

Don't know his last name: Did you ask for his last name? If so, did he refuse to give it? After you've filled out this detailed application, for him to refuse to even give you his full name would be entirely inappropriate. Maybe he's afraid of being scammed himself and is being very cautious. I'm tempted to say that a scammer could easily give a fake name rather than refusing to give any name, but scammers are often dumb about many things.

Wants a money order: Not suspicious at all. He could just be being cautious. It's easy to write someone a bad check. He likely isn't set up to take a credit card. So if he wants something that he is confident is good, that basically leaves cashier's check and money order.

Leave money order fields blank: Which fields? Any money order I've ever gotten, you have to give the bank (or whoever) the money, or they take it out of your account, and then the bank prints the amount of the money order on it. That's pretty much the point of a money order: an established institution has the money so the recipient knows that it is good. All that's left to fill in is the recipient's name and address. If he wants you to leave these blank, that's suspicious. Gets back to him not wanting to tell you his name. This could be an element of a scam. Or it could just be him being afraid to tell you his name for fear that you're going to scam him.

No lease agreement: I wouldn't be shocked if someone renting a room under very informal circumstances like this didn't think to write up a lease agreement. But if you proceed you should insist on some sort of written agreement. That would protect you from some scams, like someone taking your money and then denying he ever heard of you. Or, for that matter, from honest misunderstandings, like you thought the rent would be $X per month and the owner meant move in special $X for the first month but then more after that. Etc.

Name associated with the house don't match this person's name: That's certainly suspicious. As someone else on here mentioned, it's possible he found a house for rent, broke in one way or another, and now is pretending to be the owner to scam you. Maybe there's an innocent explanation. (Like, the house is really owned by a relative who lets him live there.) You could ask and see if he has a plausible story. If he gets mad and refuses to rent to you because you are "spying on him", fine, you're uneasy about all this anyway, right?

So my conclusion is that most of your red flags aren't really all that red ... but a couple of them are. Which leaves me in the position of being unable to offer any solid advice. There are a couple of suspicious things here that make me unwilling to say "no issue, go for it". But I don't think it's obviously a scam. It could just be someone unfamiliar with renting out a room trying to stumble through the process. I hate to say "ignore your worries" and have you end up getting scammed and blaming me, but I also hate to say "run away" and have you miss out on what could be a good opportunity.

In the end I guess I'd say, if the facts are ambiguous, go with your gut. If you're not sure, there are likely other potential places you can rent. You probably don't need this particular one.

I've been thinking of asking him the next time we meet in person to give me some sort of ID or proof that he really does own that property, though I'm sure that would seem like a weird thing for a potential tenant to ask.

Of course. That must be the normal way of renting a property. Both parties exchange their IDs, check for counterfeition, acquire photo-copies, sign a contract in writing which displays both parties' personal data, and the deposit is transferred at the time the contract is written/signed.

Precautions for the borrower

The borrower should protect themselves from scams by making triple-sure that the lender is the real owner of the house. Asking to see the original deed should be good, but it could be seen rude ("don't you trust me? do you think i am a scammer?").

I see that the OP mentioned the lock-box with a combination. I have never seen this. Usually keys are exchanged securely. If this is a consolidated habit in your territory, I must object that this must change.

One might also ask the neighbours about the identity of the landlord/lady. Or do a background check at Town Hall offices, asking who is the owner of the apartment. Even in the privacy-strictest jurisdictions, information on property is likely publicly accessible.

This doesn't prevent multiple lends, but as soon as the identity of the landperson is verified, the borrower(s) can sue them, despite being a long and expensive process covered by the property itself. Really, it shouldn't happen.

In general, RE agencies provide protection from scams. Their fee is repayed by their reputation. Then, one has to make triple sure that the agent really works for a qualified agency. Signing a contract at the RE agency, getting the keys and paying deposit on the spot is an excellent idea.

Precautions for the lender

The lender may also want to conduct background checks over the potential borrower, who could just decide to stop paying at some time, or destroy the property and be unable to repay it back (for lack of seizable value).

The number of personal data required by the borrower to the lender should be wide enough to conduct a thorough search, but must comply with privacy laws and not attempt at the lender's safety.

Disclaimer now: I am EU and of course laws and habits change. I learn that the US Social Security Number is somehow a secret, because by simply knowing it could be possible to impersonate the target. EU's national Tax Identification Numbers don't provide alone this capability, and the TIN is not that kind of confidential.

Asking for a credit report, for a work contract or the last payroll could be a solution. Of course, I learn that pulling a credit report immediately drops the credit score, which is just a disadvantage of the US credit system.