First , it sounds like your starter motor is wearing (probably where you've been trying to start it excessively recently with this problem). You're not the first to have starter issues on the supra, so don't worry to much about it, a new starter will cure the issue.
What alram system do you have. If you have a model that has a key mechanism on the alarm unit itself, where you actually insert a key and turn it to disable it completely, then this is certainly worth a try. Failing that, find out where the 'brain' part of the alarm system is and basically disconnect it. See where you stand then.
At least you know it's the fuel pump is not running for whatever reason, but the fuel pump itself works when fed a live, so it narrows it down alot.