She's absolutely gorgeous
She looks bright and perky and healthy there
As for the vet check up after 3rd to 7th day, that isn't necessary if she seems well. Unless she had some issue when you got her. When you say she doesn't eat or drink much, that's normal. We rarely see them eating. They tend to pouch food and hoard it and eat from their hoards. Their water hardly seems to go down at all, especially when younger, but she will be drinking or she wouldn't be alive! They can't live more than about 2 days without water. The important thing, if using a bottle, is to check daily that it's letting water out and hasn't stopped working - by tapping your finger over the spout. You could give her a water bowl as well, then you might see when she's drinking. For a water bowl though, it needs to be small and heavy - and in a location where she's not likely to fall in it or walk through it. Something like the little Ikea glasig tealight holders (I use those). My Syrian uses both - the bottle and the bowl. If using a bowl, the water needs changing every day.
So settling in - it takes them a few days to decide where to nest. It can take them a good two weeks to fully adjust to their new enclosure and surroundings and feel more secure and confident. So during that time, it's best to leave everything alone. You can add the odd new thing without it stressing them, as long as it doesn't mean moving things around or taking anything away generally. After a week you can just spot clean any pee - they often pee in a corner of the enclosure, or baby hamsters (which she will be) sometimes pee where they are sleeping at first, and that is tricky as you don't want to disturb her nest ideally. However the rule of thumb is - if you can't see it and can't smell it, then don't worry about it for now. It'll be fine.
Hence a vet check after 3 to 7 days will disturb her settling in time. If you do decide to do it I would leave it at least 7 days.
They groom a lot, which is normal - they are very clean little things. Also baby hamsters are quite scared at first and easily frightened, hence her flipping if you try and touch her. It's just a bit of time and patience
Hand taming is usually better done out of the cage than in it, because their enclosure/cage is their territory. They behave differently out of it. But it's best to wait till she's been with you for at least two weeks before trying out of cage time - unless she shows clear signs of wanting to come out (eg coming to the door).
She sounds fine to me. Try putting a tiny bit of cucumber out - I bet that is gone next day
You could start introducing veggies as well as her hamster mix - every 3 days for the first couple of weeks and then daily. They love fresh veg. Cucumber, broccoli, carrot are all popular. But just a tiny piece no bigger than her ear. They eat it all. Anything bigger and they can hoard it and then it goes rotten and you have to poke about in the next, which is best avoided. I've linked our cage cleaning tips below as well, as too much cleaning out, will set her back as well. There are ways of doing it to minimise stress.
Tips on how to clean your hamster cage and avoid causing stress to your hamster. It's best to have plenty of depth of substrate and mainly spot clean your hamster cage. Avoid disturbing your hamster's nest and hoard.
www.thehamsterforum.com
. There is the occasional exception but your hammy looks so innocent - usually they only bite of very frightened or in pain and even then it might just be a gentle nip to let you know they’re not comfortable.