1) Depends. Some coaches focus on technical application (with emphasis in drilling). Some in hard rolling, how you apply under pressure. Others look at how you are in the gym when relaxed. Ultimately these will overlap. Some coaches belt blues fairly quickly, and extend the wait to purple. Others have a higher expectation of a blue belt. Some times its a lineage thing ( one coach following their coaches mindset) and other times its an individual coaches own assessment based of their own journey and personality/ world view. Some its not even about rolling skill, since that isn't always in your control. Its about theoretical knowledge and coaching.

2) Nothing is stopping you. BJJ isn't like other martial arts. Although sadly its becoming that way with an influx of whiners racing on here complaining that their partner did ABC, but that's another issue. Basically in BJJ its all about rolling. You will roll and you will be found out. Immediately. If you're an easy to tap blue, white belts will laugh and talk it up amongst themselves. If you're a monster, higher belts will talk amongst themselves and will want to roll you and find out (unless they are part of the other issue I raised :S) If you went and bought a blue or purple and just went to a school you would just get smashed immediately and everyone would laugh at you. Maybe not to your face. If someone is more aggressive/ they surmise you faked it and don't have like a disability or mental issue that is impacting your rolling (if so and you put time in you still deserve your belt for your time and knowledge) they might even call you out in front of everyone. Depends really.

3) Coach is. Like I said criteria varies, but it all comes out on the mat. Obviously you haven't trained before so you don't know, but BJJ is very difference to like TQD everyone gets a black belt. Saying that there are many professional fakes at many academies. Guys that avoid rolling anyone good, always 'injured', race here with excuses and complaints if they get tapped or the opponent went too hard. Or the knowledge fraud, for example being miffed at the level of mma bjj, as to insinuate that they a blue belt could apply grappling better than a full time training fighter. There's a lot on this forum, scroll through a few threads. For me there are the outright frauds (in the scenario you presented) and professional frauds like I just mentioned. Hope this helps.