How does profit correlate with the duration of options?

The answer is more complex than you realize so there's no one size fits all answer.

The simplest answer is that nearer term options cost less so the ROI is higher if you get a move in your direction. Suppose that you are buying at-the-money calls. The extreme example is a very large move in your favor that drives all of the now in-the-money option price to their intrinsic value. Your profit will be the intrinsic less the cost of option. Cheaper options make more. However, this is far less likely to happen with one and two year LEAPs because some of that price move is converted to time premium and that reduces profitability.

Another way to look at the above is via an equal dollar investment. I could buy ten one week options for a one dollar each or I could buy one $10 same strike option that expires in one year. That's leverage. If you get that aforementioned move, the bang for the buck is with the leverage.

Here's the big fly in the ointment in your question. Part of your thesis is:

Pre-suppose you're "playing earnings with options" (the day before earnings).

When there's a pending earnings announcement, implied volatility increases, sometimes sky-rocketing for higher beta issues. That additional cost affects nearer term expirations far more than further term expirations. You therefore 'overpay' more for near expirations and that's a large drag on near term option performance.

To complicate this picture even more, with longer term options more, dollar-wise, IV contraction affects longer term options more. IOW, if you have a one week option one dollar option and a one year same strike ten dollar option, a 50% IV contraction is going to result in a loss of 50 cents and five dollars, respectively (the contraction isn't exactly 50% in each but it's close enough).

So you have nearer term options that cost more in terms of IV and further term options that are hit harder dollar-wise by IV contraction and now we have to throw an indeterminate price gain into the mix. Words can't describe all of the possible P&L's.

The short answer is that if you utilize an option pricing model and vary the inputs, you'll get a feel for the behavior of different options as price, time and implied volatility vary.

And as an aside, I'd suggest that you avoid buying expensive options the day before an earnings announcement. You're paying through the nose and you have to be more right just to break even, let alone profit. If you're going to 'play', consider ways to sell expensive near term premium and buy less expensive further expiration premium.