Visa expires today

I have been living in Taiwan for about a year and a half now. When I came, I was originally here on a work visa and teaching English. I have been studying Chinese since I came. I switched to a student visa last October when I changed my job to one that doesn’t provide a visa.
I’ve been getting the visa extension stamp every two months and am due for another one today.
The problem is, I haven’t been to class in the past month, for a variety of personal reasons I will not go into here. This means that my paperwork won’t be good enough for me to get a visa extension, right? I’m pretty sure the absences will show up and that will disqualify me.
I’d like to stay in Tawain through the end of the year, but I would like to take the summer off of Chinese classes and start again in September.
I realize this means I’ll have to leave the country for a visa, which I’m OK with doing and even looking forward to.
My question is, if my visa is due to be renewed today, May 15, is there a grace period that I have to leave the country? What will happen to me when I leave if I overstay? Will it be harder to get back in?
I currently have plane tickets to leave the country on June 13 (I planned this trip, for fun, a while ago). Is there anyway I can just stay through then, get a new visa in HK and then come back in?
Another thing I’m really worried about is that I don’t know what kind of documents you need to get a visitor visa from outside of the country. I don’t have family here and I don’t have any “real” reason for staying, at least not one that I can document. Do you need this kind of thing to stay?
Does anybody know what my options are at this point?
I realize I’ve waited far too long, but I just looked at my passport yesterday and realized I was due for renewal.

I believe a 1-10 day overstay is a fine of 1,000 NT$ which can be paid at the airport. 11-30 days is a 3,000NT$ fine and a trip to the foreign affairs police.

Call the FAP and ask. That’s the safest best.

And they may or may not know that you haven’t attended classes. They don’t ask, don’t tell.

When you overstay, even if it is only for 1 day, you automatically get a nice stamp in your passport, after you pay the $1000 NT, that says “NO VISA-EXEMPTION WILL BE GRANTED AGAIN TO THE BEARER OF THIS PASSPORT FOR ENTRY INTO ROC PRIOR TO (DAY/MONTH/YEAR)” [1 year from date stamped]. So basically you couldn’t just visit Taiwan again for a year but you’d be ok if you got a visa prior to arriving.

And you waited until the day your visa expires to make enquiries because…?