Apparently (based on information I found in a canon promotional video), all 45 of these focus points are crosstype, most of the time.

All 45 points are cross type points, but only a minority of lenses allow all 45 points to function as cross type points.

Take a look at the T7i manual (PDF), specifically the section titled Lenses and Usable AF Points starting on page 131. Canon categorizes its lenses according to how they work with the AF system, with different groups of lenses allowing functionality with different sets of AF points. Groups A through H are described, and the Lens Group Designations section starting on page 135 lists each available Canon lens and which group that lens falls into.

For example I have read that using the 10-18mm ef-s lens, only the center 15 are crosstype.

Yes, according to the manual the EF-S 10-18mm falls into group D, meaning that the 15 points in the center section work as cross type points, and the remaining points are sensitive to horizontal lines only.

In addition, will my 18-55 kit lens from 2010 have all crosstype points? Will my 24mm pancake?

There are nine different versions of the EF-S 18-55 lens listed, and I don't know which you have, so you'll need to look for yourself. Of those nine, five fall into group C (35 cross type points, 10 horizontal only), three are in group B (45 cross points, but no center double-cross point), and one is in group D. You might want to cross your fingers. ;-)

The EF-S 24mm f/2.8 STM (and all the other 24mm primes) are in group A, meaning all 45 points are cross points, and the center one is a double-cross point that's sensitive to diagonal lines as well as horizontal and vertical lines.

Will third party lenses have all crosstype points?

There's a post on the Lens Rentals blog that categorizes a list of third party lenses. However, that post was written in 2012 and relates to the Canon 5D mk III AF system, and the patterns for the various groups are somewhat different. Your best bet is to either try it, if you already have the lens, or ask the lens manufacturer.

What is it about any particular lens that determines whether all points are crosstype, or only a subset?

It seems to be a combination of focal length and aperture, judging from the list. Phase detection sensors need to receive light from a range of angles to work right. I'd guess that the AF points on the periphery don't have a wide enough field of view with some lenses to let the vertical line sensors work. If the AF points were laid out such that some of them landed near the top or bottom of the image circle, those might not work for horizontal lines with some lenses.