Hi all, thanks again for all the advice, am feeling better now. Here come some details and more questions.
The cage we have got is from Pets at Home (shown below). It has two levels, however we have removed the top level on the second day (which I now think might have been a silly thing to do but we thought it would make him more sociable as there is advice to not give them a house for example for first few days so they cannot hide). He was still able to climb into the tubes easily.
Cage setup is pretty basic at the moment (see photos). We have a few toys ready for him but decided to not overcrowd the cage at the moment. We have covered part of the cage with a towel to give him a bit of privacy (this came from the fact that the school hamster came to us with a towel covering the cage, so we thought it might be a good idea - yes we are looking after a school hamster too at the moment, and she is the friendliest hamster I have ever seen)
The hamster himself came from a litter of hamsters from a small pet shop (that was where we got our rats from so we knew it was a reasonably good place... family business and all, small breeders). He is 7 week old as of yesterday. The food is their hamster seed mix. We have given him a few treats as in bits of cucumber, apple and corn on cob. He loves the cucumber and the corn, but seems to not like the apple particularly.
As I said, first two days he was quite friendly, explored the cage, used the wheel, would poke his nose out through the door when I opened it, accept treats from my hand and let me stroke him with one finger. Apart from the incident mentioned, we did not try to pick him up at all, and he remained friendly even after my daughter picked him up.
Anyway, we will leave him alone now for at least a week following your advice. However, a few more questions (all related to the issue if we should interfere with his environment at all):
1. Should we put the shelf back or leave the cage as it is now?
2. Should we put more toys in? We have a little wooden climbing "block", a crunchy tunnel, and my daughter could make him a hammock - the school hamster loves the one she made for her.
3. We also have seed and fruit sticks. Would it be a good idea to hang them for the new hamster?
4. Speaking about the toilet - we tried to toilet train the school hamster. Bought her a toilet and added some soiled bedding there, and keep doing it every time we clean the cage. She absolutely loves her toilet. Only she uses it to groom herself and to hide food there
Have not seen her use it as a toilet at all, she still wees mostly in the tiniest corner behind the house and poos everywhere. We also have a toiled for the new hamster but were waiting to see where he chooses to do his business so we can set up toilet area there. But since he seems to be doing it in his house I do not see a point in putting the toilet in, or am I wrong?
PS Before you ask - we all do wash our hands very carefully after we handle each hamster.
PPS Another quick question - I forgot that we have alarm sensors in all rooms which will glow blue when there is movement. I am wondering if I should cover it up, it may be a bit freaky in the night if it reacts to hamster moving... Need to somehow check if it does actually