Re the tyrers,
A) yes
B) No
C) won't make any difference, you'll be wet from the pool and still get left behind by 80% of the field.
But it doesn't matter. Because first off tri isn't like a run race where anyone is snooty about who comes in laater on in the day, and because even with £6000 of carbon tri bike you'll still make enough other bloopers in your first race to still be off the pace. Just do your first one to get a feel for the process, learn transitions and get an idea of pace.
In the medium term slick MTB tyres will be a good idea, and you can instantly become semicompetitive on the bike by doing that. But there will be other people on MTBs, so don't worry about it too much. If it's a pool swim, and you're swimming isn't Phelps like then you'll be off early, and so get a 2 hour head start on the the fishes, so you'll not be out on teh course on your own anyway.
Shoes, almost everyone changes shoes, mainly as most people would be using bike shoes that clip into the pedals
Shorts - the only rule of race 1 is not to get naked, that's a big no no. So you really should consider spending £30 on some tri shorts that are designed to swim / bike /run and won't rub your nads raw. You can just about manage for a sprint wihout, but it's not going to be fun.
Water, entirely up to you. For a sprint then I tend to take a 500ml on the bike and only drink half, but it's your call.
Other things, then don't pin your numers all the way through your top, it happens every race I referee! Put your top on, then pin the numbers on, or get someone (you trust) to do it for you. Then roll up your t shirt from the bottom as you take it off into a donut so that when you're wet and you put it on then it sticks and rolls down. If you eave it flat then as you're wet the bottom sticks on your shoulders and it all goes pear shaped. If you want to see what I mean then try putting a t-shirt/vest on just after a shower before you get dried.
Make sure your helmet is stickered with 'EN1078' and make sure you put that on before you touch your bike, and then rack your bike again before touching your helmet. Not even unclipping the chin strap.
Transition box. Don't take too much stuff. Classic newbiw mistake. First, your car may be huge, but your space in transition won't be. Second, then when you're in mid race then you really just want the essentials there. So top, bike shoes, helmet, glasses, run shoes and possibly run bottle. Keep the rest in the car, including your pre and post race clothes. The more that's there, the more that you;ll end up faffing with in mid race. And no walkman/i-pods at any time, even pre-race in transition.