The second genitive case is so called partitive case. Not all of nouns can be used in second genitive case, usually short ones or/and uncountable, like sugar, tea. It is not obliged to use this alternative form, in exception of words combinations like ни разу, выпить чайку*, and phraseological combination без году неделя; с миру по нитке; моя хата с краю; беситься с жиру, без роду без племени and so on. I have never heard дому, but I think it is possible to say без дому in spoken language. * It is better to use the second genitive with diminutive uncountable nouns. Кофеёк, чаёк, сахарок. Выпей кофейку, чайку, насыпь сахарку. But as usually using diminutive nouns is more often in women language, you need not remember it.

There are approximately 500 masculine nouns that have two genitive singular forms -- а(я) and у(ю). There are basically interchangeble and their usage is just a matter of style. You can use the first one (а/я). There are two different singular endings in the prepositional case (for masculine nouns) -- е (the main one) and у(ю) and several in genetive plural (all genders) -- ов, ев, ей, ен, ель. In these cases you should memorize which one to use.