As long as your form is good, then you are more unlikely to injure yourself.
For rows, some 'body English' is fine to help drive progressions, especially since it is not a competitive movement, so being strict with momentum isn't inherently necessary.
If doing Pendlay rows (as advocated by Mehdi), you can have a bit of thoracic extension to aid with getting the bar off the ground. Controlling the eccentric portion (the decent) of the lift will help with strength gains as well.
Glenn Pendlay Teaches the Pendlay Row
Alan Thrall's "How To" Barbell Row
For normal bent-over-rows, since you don't reset the bar between reps, there will be more emphasis on the eccentric portion of the lift and you can make use of a stretch reflex at the bottom of the eccentric which will help following reps.
Scott Herman's How To: Barbell Bent-Over-Row
As for presses, those are a bit more technical (more-so than a lot of beginners believe initially). Improving form on the press can usually help with progressing again, but not always. You also have to consider that the press utilizes a smaller set of muscles than the other common compound movements. It is very common to see slow progression with the press. For instance, it recently took my about a month and a half to put only 10 lbs on my press.
Presses can also benefit from a variety in the rep ranges that you use to train it. One training session may involve a max effort 3x5 barbell press, and then the next session may involve a 3x8 dumbbell press for hypertrophy. It may take some time to find what works well for you.
Brett Contreras: How to Military Press
Don't worry about the trainers, as long as your form is correct. Proper nutrition, rest, and technique will all help you progress in the gym.
Squats are literally a full body exercise; they involve a lot of muscles. It is very common for them to feel taxing, which is why eventually a lot of people transition to squatting with less frequency per week (which you don't need to worry about just yet). However, if you are finding recovery too difficulty with 5 straight sets, you can try squatting 3x5 instead (which is actually done in Starting Strength). But you need to be honest with yourself and not go that route simply because it seems easier, but because you're having issues with recovery. I'd recommend sticking with the program until you plateau 2-3 times on either your squat or deadlift and then move to a more intermediate program.
It's hard to say without more information, but you may also find an increase in caloric intake to be helpful. A caloric surplus will help with building muscle, which in turn will help with your strength gains.