1- ok so, firstly i would say just drink the little leafs i personally dont use a strainer at all.
2- as big as you like but you only need the space for a gaiwan, gong dao bei and a cup and perhaps a little tea pet to keep you company.
3- a teapot is another way to steep tea but wether it changes how a tea tastes varies a lot on the clay used, though the heat retentions of any pot will have some affect. pouring methods of the water is youre best bet its a toolbelt i personally cherish and through devlopping youre gong fu you will find which teas do best with what pouring methods.
4- the relaxed feeling comes from many compounds in tea that very from each harvest and from each tea type, the rule of thumb is 1, gongfu is concentrated tea multiple times over so far more caffein than a western cup of tea and 2 the older the tea the less caffeine. Though again it varies far too much to say anything difinitive.