Hello and welcome
All sounds like normal behaviour! In particular they pouched their food and hoard it (often under their nest/bed) and snack from the hoard. Climbing and jumping sounds ok as long as the cage isn't too tall and there's nothing hard or sharp he could land on or bounce off on the way down.
Best advice I could give now is don't clean anything or change anything for the first two weeks while he's settling in and scent marking his way round the cage, to avoid stress and setbacks. Also using a litter tray. Syrian hamsters will use a toilet/litter tray for peeing in and it helps keep the cage clean so less clean outs (which stress them) are necessary. A dish with some chinchilla bathing sand in is ideal for a litter tray or you can buy corner plastic litter trays (they often choose a corner of the cage as a toilet area for peeing). The main thing is to put it in the right place. If you put it in the chosen pee spot they will use it - if you put it anywhere else they won't!
If the cage gets a bit whiffy in the first two weeks you can "spot clean" maybe once a week during that time (ie take out a handful of wet soiled litter and replace it with a handful of clean and mix it in a bit so it still smells familiar. That's an ideal time to add a litter tray.
I've used these which are a good size - not too big not too small. They are quite clean and tidy and tend to pee in the very back corner (where it's higher). Ours have then sat in the front bit for a wash.
https://www.amazon.co.uk/Hamster-cor...=Hamster+potty
This is the Chinchilla Bathing sand a lot of people use (must be sand not "dust"). One of these lasts me a few months and I empty out the potty every 5 days or so.
https://www.amazon.co.uk/Supreme-Pet...a+bathing+sand
He sounds to be having fun getting his cage just how he wants everything
Using a litter tray, spot cleaning and having a good depth of substrate, you can easily go a couple of months without needing to do a big cage clean and even then "partial" cleans are better to avoid stress. So do the litter/substrate one week, wheel a different week and any toys etc a different time again - so something always smells familiar.
I also replace some of the old clean litter and mix it into the new or sprinkle it on top so it smells familiar. Also best not to throw away their nest or hoard. Those are their two most important things in life and help them feel secure. There is no need to unless they are pee'd on. (They do have slips when they're young!). If they are pee'd on then try and leave behind a dry bit of the old nest and replace any soiled hoard you remove with new food in exactly the same place. Dry hoard can be left for a while - eventually it starts taking over the bottom of the cage and needs pruning at the next substrate change.
Dont worry about poops - they're like little hard black seeds and not smelly or dirty - and your hamster may eat some of them (normal for extra vitamins) or hoard them (supplies in case food is every short). So again unless they start taking over in a big way don't worry about leaving them.
Assume you know about wheels and wheel sizes. Most people get an 11" to 12" wheel for a Syrian. An 8 to 10 inch may be ok but some grow out of them and they tend to run better in a larger wheel. Anything less than 8" diameter is too small - they need to run with a straight back, not curved, to avoid pain and spinal problems as they get older.