Welcome Bannoy

You are smart to do everything "by the book."

It seems like you are a complete beginner? And that is great - always wonderful to have new entries.

Recommended bike? That is a completely personal decision. But as a first bike, I'd recommend a small displacement between 100 and 150cc. A 250 may be too powerful, at first. The reality of Beijing traffic is that you are not going to go anywhere fast. A light, narrow & nimble bike is best.

Your budget is a little tight for a new, legal bike. I think you might be able to swing a new Honda SDH-150 with a "B" plate for under 10K.

I've recently seen used 150cc sport-type bikes (Chinese made) on The Beijinger for 8K, but I don't know anything about those bikes. Be careful about what "complete with papers" means. Personally, I don't think that a sport bike is appropriate for a new rider. The usual recommendation for new riders is a more upright seating position.

Insurance is cheap - only a few hundred rmb/year. But it only covers 3rd party liability. It is required. Any PICC office can sell it to you.

Advice? Keep it simple. Take it slow. Find a more experienced rider, explain that you are a new rider, and ride behind them, watching what they are doing and ask LOTS of questions. ASK about everything. If the "experienced rider" appears to be taking risks, find someone else. NEVER let someone push you beyond your abilities. Truly experienced riders always ride at the level of the least experienced in a group.

ALWAYS wear a helmet and protective gear. You can control your bike but you cannot control what other drivers are doing.

Riding safely is riding with enjoyment.

OK, sermon is over.

PS: I have to say that learning MC riding in Beijing is a little bit of trial-by-fire. Good luck!