Short answer: the difference is in an SLR the mirror flips up to take a photo, whereas the SLT employs a semi-transparent mirror so that the mirror never has to move.

Now to your questions:

1) How does a DSLR do continuous focusing (AI servo/AFC)? Does it literally flip the mirror up, take the shot, put the mirror down, refocuses and repeat?

That's correct, assuming the DSLR is not in live-view mode.

If the camera has a live-view mode and that is turned on, then the mirror will stay up the whole time (and you won't see anything through the optical viewfinder at any time) and it'll switch to CDAF which is usually slower in a DSLR.

Wheras in the SLT camera it should be able to continuously use PDAF auto-focus while taking a shot or not.

2) How does a DSLR user account for the fact that the OVF has significantly greater dynamic range than the actual user? Do you just get used to it and compensate mentally?

Yes, you get used to it, just as you get used to judging the contrast of a scene with just your eyes - no camera at all.

Besides, even with cameras that have a live-view or digital viewfinder, you can't rely on the depth of blacks or contrast shown there to be indicative of the recorded picture - often sunlight, reflections and the relative quality of the LCD display make that hard. You still have to do some level of mental correction. There's also histograms and things that you can turn on.