I'm really not sure of the correct grammatical use, but in everyday situations with or without the "that" would sound just fine. I believe that in Spanish it is more correct (or sounds much better) not to miss the correct useage of "that". Sadly, in English incorrect grammar useage is very common and it is hard sometimes to remember how it is supposed to go. :P

Both are fine.

"that" can be left out or included in both sentences. Google "relative pronoun omission" for some more information on this.