Real Estate Law - Making offers on a home?

No, the banks take highest AND best, not just highest.

The more cash the better. They also may have processed the other offer before yours was in.

There is nothing illegal about what they did. They do not have to respond to your offer in any way.

Probably not. There are many factors considered when choosing among multiple offers. Price is always important, but so are financing terms, contingencies, etc. Your $360k with a pre-approval letter from your lender may not beat an offer of $330k all cash (or no financing contingency or something else).

Do you have reason to suspect that the listing agent didn't present your offer? That would be fraud, and that would be a very big deal, of course. I would ask the listing agent why your offer was passed over. If no response is given, you could ask the LA's broker what happened. Again, I'd only do this if I had reason to suspect fraud.

Sellers are under no obligation to accept the highest dollar value offer. There may have been other components of your offer which the seller found undesirable.

My guess is that the bank didn't look favorably at your offer because they fully realize you're not going to get financing for $460K on a house listed for $340K. No appraiser is THAT far off the mark.

No legal recourse. The only question is why the bank made such a poor decision. I asked a listing agent once. She told me she was faxing off 6 offers to the bank. The best offer was the last in the pile. When she was faxing offer #3, the bank called her and said it had accepted offer #2. She said that the best offer had not even gotten in the fax. The banker said, "Offer #2 meets our conditions". She was completely shocked. .

The seller, in this case the bank, can accept whatever offer they want to accept. They accepted the other offer instead of yours. They have the right to do that and they don't have to give a reason. It could be that the other offer had come in and been accepted before your offer was presented. If they had already signed the contract on the other offer then they couldn't even entertain your offer. They didn't do anything illegal.

They only way you would have a case is if you feel you were discriminated against for being a certain race or sexuality.

A big decision for the bank could have been someone was to offer cash vs you needing a loan.

No you don't, unless you can prove there was some sort of discrimination. It is possible the lender knew the home wasn't going to appraise for your offer price, making your offer problematic. Lenders obtain appraisals and broker's price opinions to determine their asking price, so they know what it is going to appraise for.