Imo, it is better to give priority to getting vaccination. In taipei, most doctors speak English more or less. Just show them the immunization record (what I have is a yellow card at the first page of the handbook) with your kids. They may understand what you want. You can use Google translator to show them in Chinese what you want to tell.
Pediatricians do the well baby exam and vaccination. They often accept adults patients too. For adults no regular chuckup. For 2 yr old, maybe once or twice a year, for 3, once a year, and for 4 in not sure now but might be none.
If you have not read yet, this topic may help some.
Well child exam schedule in Taiwan
I put there a link to a booklet in which pp19–24 are on immunization. The booklet is written in both languages, so might help you get some clue to know what is written in your children’s handbook.
as for the 3 days part. That may be because more than 3 days are not covered by NHI. Didn’t doctor say to come back after 3 days? If you pay by yourself for additional 4 days, doctor would prescribe 7 days. Its cost is about 100NTD. But doctors might prefer to follow up after 3 days.