Translation:I will have given the cat a bath when my mother sleeps.
Is "em" really the preposition that is used to say "give a bath TO"? Is "a" also a preposition that can be used?
no. we say "dar banho em alguém/alguma coisa" (Vou dar banho no meu avô)
Yes, you would use "a" in Portugal, instead of "em". For example, "vai dar banho ao cão" is a common expression. See here a nice post only laterally related to it: http://opiniana.blogspot.co.uk/2013/10/vai-dar-banho-ao-cao.html
That is one of the differences between the European and the Brasilian standards for Portuguese. In BP you say "no gato". In the European standard, which I think is also followed in Africa, you say "ao gato".
Again, I feel the need to correct this because in English we would use "while" to indicate simultaneous actions taking place.
"I will have given the cat a bath while my mother is asleep."
Agreed. 'When' is incorrect and should be 'while'. 'Is asleep' could also be 'sleeps' or 'slept', 'slumbers' etc. All mean essentially, if not exactly, the same thing.
It calls "futuro do subjuntivo". You'll see more about it through the lessons.
OK, if it is "banho no gato" etc. in Brazil, then what is the idea, looking into the semantics. Is it an inversion maybe of what in English would be expressed, "Pour the bathwater on the baby?" That must be it......From the good old days when pitchers were used.....
Okay try to bear with me here: I'm wondering about the time of the verb "sleep" in the English translation.
If I understand the sentence correctly, the author of the sentence is describing a future action that will have already happened: in the future, my mother will sleep, during which time I will give the cat a bath, and the sentence is expressing a point of view from a moment after that bath has been given. Therefore, wouldn't the mother's sleep also logically be in the past from that future point of view? Wouldn't "I will have given the cat a bath while my mother slept" be a better fit?
Or does the portuguese sentence really say that at the moment in the future the sentence is describing, the bath-giving will be past, but the mother will be asleep? Or are both possible?
The problem is that "sleep" is a continuous activity, so it is very possible that she will still be asleep when the other activity has been completed (and is thus in the past). So yes, both are possible, you use "slept" when she will have woken up by whenever you're talking about, but if she will still be asleep, or you don't know if she will still be asleep, you use "sleeps"