Well, programs like Stronglifts and SS are really meant to be done to the T. Anything less won't produce the same results, anything more might impede recovery and hurt progress. But you are going to have to modify this program to make real progress.
I would do the pushups on the Stronglifts days, but drop the number of sets in both the main lifts and pushups. Do bench and OH press for 3X5 instead of 5X5, and follow that up with 5 sets of 30 pushups, or something like that.
Definitely don't do the pushups super intensely on the rest days. You could do a few, but that will really kill your progress in the weight room. (At least it does for me)
Also, make sure you keep the rest short in between the pushups. You should be resting about 3 min in between sets for Stronglifts, but for these pushups, I would go for 1 minute or less of rest.
For the marches, I would suggest some kind of loaded walk to get used to walking with a weight. I would get a weight on your beck in a squat position (go for high bar because you won't actually be squatting and it'll be more comfortable) and walk with it. You'll learn how to walk correctly with weight and not hurt yourself. Also, If you use more than 50 lbs, you'll be used to the weight so the 50 won't feel quite so bad.
I would not use any kind of aid that is not going to be accessible to you at training. Train as though you will be given no assistance so you can be ready if you aren't. If they do give you something, take it as a nice surprise.