Gostar always has to be used with "de" in the present tense. As for the "the", as in o seu pente, I think it is optional. Like so helpfully explained by @Paulenrique above, when de precedes any definite article, it combines into do da dos or das. So basically, as far as I understand, you could say Eu gosto de seu pente, or Eu gosto do seu pente. Either way, you have to have the de. Hope this helps!
For a few reasons, yes! It is best to remember the verb as 'gostar de' (to be fond OF, if you like), which gives us the start of your sentence "Eu gosto de .." ; at the end we need to use 'sua' in place of 'seu' to agree with the feminine noun 'a casa'; then in the middle using Paulenrique's advice, we pick 'da' as short for 'de + a' to end with "Eu gosto da sua casa". Hope this helps!
So this is not an obscure way to say "Wow, what have you done with your hair?"