If I were beginning a profile as a bounty hunter would it be possible for my character to pick up a rather crappy blaster off of a guy who got killed in a brawl simply for the purpose of basic protection on his way to mandalore?

yes, though when you become a bounty hunter, you will get a basic blaster for training, no one in that profession would be unarmed. When you progress you can eitiher buy new weapons and weapons will of course as a mandalorian be attached to your armour which is in your case free when you graduate.

what exactly are the basic weapons, is it any weapon under 500 credits?

Well basic weapons are weapons which can't kill only hurt. Sometimes for harder missions you will get more damaging weapons. However now with the new rule, you can immediatley get the armour with all its weapons attached, now that we decided to skip training. You can immediatly proceed to missions if you want to. But there are higher levels you can reach as a bounty hunter. For example if you want to become a Jedi hunter, you'll have to stay on mandalore for training and you'll get a lightsaber to train with. If you prove yourself to be a good bounty hunter you can also become an instructor.