Is it common for dentists to not prescribe pain meds?
In my experience yes that is uncommon. Typically they prescribe either a more concentrated ibuprofin dosage (plus a small bit of narcotic IIRC?) or just a narcotic like loratab. I disliked that personally given the addiction potential, they way overprescibed. Eventually I just started tossing the scripts... You should be fine with OTC ibuprofin or booze. I've had sockets with zero pain.
And yes it's healthy, IIRC do not remove it! Read up on socket healing. High correlation between leaving that initial thick blood there and a better heal and shallow socket. By cleaning it out you will get a permanent deeper socket. Don't brush it out when brushing teeth, don't suck on straws, and don't rinse forcefully, let it slide/drool out. Then again it is your mouth and you have the freedom to do whatever you want. Just be aware a deeper socket is considered worse (how it is worse I don't know).