Wow! Number one welcome
secondly, when it comes to gates, practise practise practise, it's the only way you'll get used to riding in a mob. Other games like knock-off or 'foot down' (I think they call it here) will help deal with that fear as well. Just because you knock against someone doesn't mean that you or they will fall off! If you are finding you're slow off the gate take a look at your techniques and ask a coach if you're doing it right? There are a few little adjustments you can make, that will make a world of difference.
Race uniform, nah. I know pro's who ride in trackies some days, ride with whatever you feel comfortable in (provided it's UCI approved) Bear in mind that race gear, mostly pants, will provide a bit more protection when you bite the dust but it's all relative.
Chest protection?? I assume you're talking about body amour... The plastic stuff was only ever designed for moto roost and doesn't really do crap all on a BMX track except save a tiny bit of skin (they tend to have stickers on them saying, this stuff will not stop you from getting hurt). Otherwise, protection like the 661 Pressure Suits can be great for building that confidence, a lot lighter and easier to move in. But they're not for everyone, give it a go and see what you think? Failing that you can always work the elbows and knees combo
Hope this helps!