In the kneeling hamstring stretch, I feel a lot of pressure/stretch behind the front leg's knee. Should I ease up a bit?

This is very common. People have a ton of tightness there. People got tight hamstrings and even tighter calves and those muscles attach behind and across the knee joint, so the sensations are going to be really strong there. You're going to have to take your time with this and regulate the position yourself. You could keep your knee bent and still feel the stretch in the belly of the hamstring rather than behind the knee, but you're probably going to want to straighten it over time as you find comfort in it. Believe it or not, the sensation goes away dramatically there the more you stretch and stretching becomes... gasp... pleasurable!

After the kneeling hamstring stretch, it says to straighten both legs but your rear leg's top of foot is supposed to be on the floor already. Wat do? If I straighten the back leg with the "shoelaces" already on the floor, doesn't that mean I'm on the point of my toes (which seems wrong since lots of pressure on just the toes)?

Okay so, it sounds like, you mean you're going for the front splits. When he asks you to raise the back knee off the floor, tuck the back toes under so that yes, you are on your toes. (Which is very hard btw, definitely use chairs by your sides to hold onto when trying this first.)

When doing the kneeling lunge, I feel the stretch on either side of the groin area (I think it's both legs' hip flexors?). Is this right?

Yes you're supposed to be feeling the stretching of the hip flexors (quads/psoas) for the back leg.

For side splits, what does this mean exactly? "Make sure you tilt your pelvis backward (but keep the torso upright)" Does that mean if we were to take a picture from the side, my entire body is in a vertical plane?

Okay, so I know that shit is confusing, but I think in this situation, /u/phrakture means to perform anterior pelvic tilt, meaning to stick your butt out.

For side splits, do the soles of my feet have to be on the floor, or just the inside edge of foot is fine?

Ideally, the lateral parts of the foot are down as well and the inner arch is actually lifted (like as you could stick a spatula underneath the medial arch). But if you start getting super low, you'd eventually want to actually raise your toes up and be on your heels as that externally rotates the hips and helps you open up even more. (Which makes it a lot harder to balance as well, so a chair to hold onto lightly would be nice too, again.)