So I decided to switch my multiple-entry visa to an ARC. Kind of regret it now, but what the heck. I turned in my arc application 2 or 3 days before the 60 days ran out on when I had to leave the country. I got the ARC like a week and a half later (about 5 days after my last visa thingy expired). Does this mean I overstayed and need to pay a fine? Where does one go to pay these fines? I don’t want to show up at the airport on the way home in a few weeks and get stuck in some visa mess. Do they take into account the lag time between application and approval? Will this negatively affect me in the future?
I did that once when getting an ARC for work. I didn’t have to pay a fine, but got a “naughty boy” stamp in my passport. This caused me a lot of hassle in HK, but when I explained to the lady at HK immigration that I was in HK to pick up a Taiwan work permit she said “I’ve never heard anything like it” wrote her report and I was on my way. It was MOFA that messed me up and I had to do a lot of too-ing and fro-ing. But no fine. I got my extension/slap on the wrist at the foreign affairs police in Taipei.
The start date of your ARC and multiple re-entry permit sticker should have been backdated by the police to the last valid day of your previous visa. Check again.
nope. nobody bothered to count 60 days from my last entry stamp. the visa wasn’t expiring, i was just near the end of the 60 days when i had to leave the country again. hopefully the people at immigration are just as lazy and don’t bother. 
Then don’t worry about it. If you have a valid ARC, then your re-entry permit is valid. And if that’s valid, then you can come and go as you please.