Yes it's good. I use a variation of that exercise very often. It is actually most commonly called a "face pull" and there are a few different ways of doing it. I would change the angle on the exercise, but it may just be the equipment he has to work with at that gym. Ideally I would start with an adjustable cable--one where you can lower or raise the pulley to different heights. I would set it about shoulder height, then do the exercise as illustrated. The focus would NOT be on the arms moving the weight but on the shoulder blades pinching together without shrugging up. The arms moving would be the "finishing" part of the exercise. While that is the easiest thing to see in a video, the real set-up work is done in the pinch at the very start of the move.
That's actually true of all rows, bent barbell, dumbbell, cable and all.
You should feel it right between your shoulder blades. You can also feel it in the back part of your shoulder (the rear deltoid referenced in the video below).
Here is an excellent video that goes over how I would generally use the face pull (except reps can vary more than he says). https://www.youtube.com/watch?v=l24pXqUrnUk
I do think that he doesn't put enough emphasis on the contraction of the mid back between the shoulder blades in his verbal explanation, but it is extremely thorough overall.
NOTE: I want to point out a couple very important things in this video--the first being, look at where his arms finish at. His elbows are NOT behind his shoulders!! They are directly to the sides, but not behind. You can actually pull the rope too far back in this move, so stop when your elbows are even with your shoulders out to the sides, or even a half inch in front of your shoulders. Don't pull your elbows behind you. 2nd, notice his posture--he is not arching his back at all. He has a straight line from his hips to his head, and even though you can't see it his abs are turned on and squeezed, as if he was doing a plank as hard as he could. This is excellent posture. The only thing that really happens is his body is leaned back a little to balance itself against the rope, which is perfectly fine.
Ok, Lat raises. I would not do the lat raises. It's not a rotator cuff exercise as much as a shoulder exercise. It's useful and I use it quite often, but if someone is already dealing with an injury or a problem like yours it can keep things from improving. Once someone is healthy again it is quite a good exercise to put in.
I would switch to dumbbells now. And I would change to a neutral grip (also called a hammer grip). If you're unfamiliar with that it basically means having your palms facing each other and your elbows tucked close to your sides instead of out wide.
Having elbows out very wide in a bench press--barbell or dumbbell--is in fact one of the things that can easily lead to shoulder issues like you are experiencing. It doesn't do it in everybody, but in people who are predisposed or at-risk, that is a big aggravating factor.
To understand why I said these things, lets go back to the lateral raise where you feel clicking and pain. Now do one of those lateral raises (without weights please) and hold the top position of the exercise. Holding your arms like that, go lie down on your back--what does that position remind you of? The middle of a barbell bench press exercise. You see how similar those positions are?
RE: foam rolling--use a softball or baseball for the shoulder areas! Foam rollers are easier to reach the legs. You can lean against a wall and lean on the baseball. Make sure you stay on muscles not bones, but otherwise good. Don't put the ball under your shoulder blades though, keep it between them or on the muscles on top of them, or on the back/side shoulder muscles, even under the shoulder near the armpit. A bit awkward to control the ball and it will bounce away from you, but once you get the hang of using it the ball is great.
Lacrosse balls are a good choice--get them from a sports store or online, they're the most common tool to "foam roll" the upper back, shoulder, and arms/forearms. The rubber sticks well to most surfaces and doesn't slide away, and the ball is hard enough to get good pressure on, not so squishy like a tennis ball.
I would do the following:
Low trap raises ("10-2 raises", also called "Y raises")
as a warm up before your lifting 2-3 times a week, about 3 sets of 10-15 reps each. Reps aren't as important as control and posture.
Wall slides here from Eric Cressey:
NOTE: when Eric says "Round" in the 1st video he doesn't mean "slouch"! Stay mostly straight haha
NOTE 2: Wall slides do not need to be done with a roller. Roller can help you feel the posture a bit because you need a little pressure on it to keep it from falling down, BUT a wall slide can just be your arms right on the wall sliding up.
Low trap raise explained well here---but poorly executed because the guy is pretty inflexible and has to bend his elbows. The idea is correct but you need your elbows straight instead of bent....the reason for not bending the elbows comes in the 2nd video which explains a very similar movement.
good explanation bad execution https://www.youtube.com/watch?v=37Z6aO8KsuI
Cressey again explaining a related exercise: https://www.youtube.com/watch?v=YbqcB5dBBD0
Note that Eric is explaining an identical exercise to the 1st video, just using a different form of resistance than dumbbells. It's harder as well. So start with the dumbbell version, but with just your arms and no weight, then graduate to weight very slowly. Forget the TRX because it is harder--but pay attention to his explanation because it is true for both exercises. Posture is everything here. Eric's posture cues work extremely well for both dumbbell and suspension trainer versions.
You may have heard the term "shoulder depression" or "depressed at the shoulder girdle" in one of these videos. That is something that can play into ulnar nerve compression. If the shoulders sit down really far (and/or are downwardly rotated, meaning rounded shoulders) they can be sitting so low or rotated so much that they pinch your ulnar nerve.