Disclaimer: I wasn't sure if to post this on Halacha, or Jewish Fundamentals or what, and I came accross it while doing the Daf, so I put it in here. If it's a mistake, please move it. Thanks.

Rabbi Meir holds that night is a time for Tephilin.
But if that is the case, then according to him, it's not a Mitzva that haZman Grama;
In which case, according to him, women would not only be allowed to put on Tephilin, but are Chayav???
Did I comepletely misunderstand this part of the Daf, or miss something in it?

If what I said is true, then how did we go from that, to today? (R' Meir wasn't a small fry...)

I'm pretty sure you are correct: there is a tanna that holds women are chayav in tefillin.
There is a tosfos I learned recently that discusses it, but I can't remember where. It is also likely that I am confusing tfillin with tzitzis.

also, we don't wear tfillin on shabbos.

Correct, Shabbos, Yom Tov and maybe Chol ha Moed are pattur from Tefillin. So it is hazman Gramma because it has to be Chol to be chayyav in teffilin.

Regarding Rashi's daughter's wearing tefillin, see here:

http://www.hashkafah.com/index.php?showtopic=1638

So really, about his daughters... we just... don't know?

Also, since there was an opinion that they're Chayav in Tephillin, are all the other opinions that women are allowed but not obligated, or is it assur for them?

In the linked topic, someone quoted a shiur, that since it is so difficult to keep a Guf Naki, women should not wear tefillin. Men have the same problem, but since they are chayav, they have no choice but to risk it.

In the time of the gemara they were better at keeping a guf naki then we are.