Eczema on the hands?

I think there is a very subtle difference. hand and hand just refers to two hands not necessarily together.hand in hand implies that the hands are connected or together.

That I believe is clasping.

hand in hand is when two people are holding each others hands tight.

Yes it can.

You use the hand that you write, eat, and do most things with. That is your strongest hand, therefore the best hand to shake someone elses hand with.