Why did the price of crude oil drop below zero today?

Can I invest in crude contracts through a traditional brokerage firm? Will I get paid for every share I purchase?

That's not how futures work - you are entering into a contract to buy crude at a certain price. When that price is negative, it means that you get paid to take crude when the contract expires. The problem is that you have to store it somewhere, and there's currently no storage available (which is why prices are negative). But yes, if your brokerage allows you to enter into futures contracts you can do so. Then you have to hope that the price of crude goes up tomorrow and you can sell it back (otherwise you'll need to find a way to take delivery of 1,000 barrels of crude per contract in Cushing, Oklahoma).

Why did the price of crude contracts drop below zero today?

Long story short, holders of futures contracts that expire tomorrow needed to sell their contracts or take delivery of crude oil with nowhere to store it. Since no one wants to buy these futures contracts, sellers were willing to pay someone to take these contracts and avoid having to take delivery of the oil.

If crude oil contracts were to continue to drop, let's say -$90 a barrel, would consumers get paid to fill up their automobiles?

Theoretically, yes, if refineries get paid $90/bbl for taking crude oil, spend money to refine and transport it, then they could still pay consumers to take gasoline and still make a tidy profit. More realistically, though, is that consumers will still be willing to pay for gasoline, so barring a price war that turns retail gasoline negative, refiners' and/or retailers' profits will skyrocket.

This has never happened before with oil, but has it happened before with other resources?

Negative interest rates (banks paying money to store their cash with the central bank) have been around for a while. I don't know of any other physical commodity that has had negative prices, though.