How dangerous are my worn rims?
Rims are worn out when the groove disappears. If the groove is clearly visible, well-defined and of uniform depth all the way around, the rim is not quite worn out and certainly not dangerous.

New brake blocks will cause less wear as they will be free of hard particles of grit and metal that get embedded in the relatively soft rubber over time.

Given the repair shop's bad advice on the state of the rims and refusal to replace the blocks, I'd disregard all of their other advice about needed repairs. I suspect they just saw a well-used bike and decided they could persuade you to purchase a new one. Find a repair shop with a better reputation and get a second opinion.

Replacing and adjusting brake blocks yourself is relatively easy if you have basic tools. Park Tool has an excellent web page and collection of videos that walk you through the process.