Even if he seemed to be picking he will eat from his hoards mostly so it’s probably fine. They often leave the log things which are the fibre bit I think so not that essential.
Yes just a idea sized amount of veg each day (or no bigger than their ear approx) or I sometimes say 1cm cubed size (which is probably slightly bigger than an ear but easier to imagine size wise)
On Harry Hamster he will already be getting enough protein - an occasional nut twice a week would Bolster it - walnut halves are popular. Any nuts need to be plain and unsalted obviously.
For an immune booster you could give a pinch of hemp seeds - they are supposed to contain all the vitamins and minerals and they are also high calorie so will help with improving his weight (nuts are also high calorie). Hamsters tend to like them and just lick them up- they smell/taste nutty I think. A bag will probably last years as you only need a “pinch” daily. It’s best to put them separate from the main food - eg on a jar lid as a dish - they just lick them all up and you can see they’ve gone. With the walnut half I put it in another part of the cage - maybe on a shelf or on the substrate - so they enjoy finding it and then that becomes a regular spot to look for it so they look forward to it
I actually do an “extra” like that each day - as they love expecting it in the same place - so twice a week it could be half a walnut, in between that it could be three or four scattered pumpkin seeds (you can use the normal green ones or if you get some of the white ones they love cracking the white shell open). Or two or three sunflower seeds (they also can increase weight ) or sometimes a “complete crock “ treat - linked below.
The apple, berry, carrot and cheese ones are popular here - I don’t get the herb ones. They aren’t sugary and don’t have added sugar.
Also does he have a chew stick? They all seem to love the mini toothbrush whimzee chews - one if those lasts a while and you can stick sunflower seeds on the bristles so they have fun getting them out
I occasionally give one of the Vitastix chew sticks as well - the honey and spelt one - the honey is sweet hence they just get one or two a year but they tend to last a while - not as bad as sugar though. Although some hamsters will strip it in one night! But then hoard it all - it’s more foraging than eating. It’s still good for their teeth chewing the hard wood stick with food on.
Amazon
For an extra special occasional treat you can give a dried sunflower head - they can pick out all the sunflower seeds and they tend to go crazy for them - hence best to only give haif if it’s a large one! Higgins had one (see his thread) the other night and completely stripped it! But they just hoard them all and snack from their hoards - one reason not to remove their entire hoard of at all possible or that makes them very anxious.
How do you go with spot cleaning? Does he pee in a toilet at all?
I get the sunflower heads from here - their products are always good quality (I had a mouldy one from somewhere else!).
They also do good sprays and I get the pumpkin seeds in their shells here as well
Hamsters we love
I am very much hoping that it was an infection and that it is gone. If something was impacting on his breathing, I would have thought that it would still be doing so, even with the steroid. The more active he is the more positive things look. I really hope he will have a good life free of illness now 
. He was soooooo much better. Like 100% fine. He's still getting the steroids, but if honestly considered stopping them as I couldn't see that there was anything the matter. I don't see how steroids could help a tumor to the extent he seemed fine.