A couple of questions(spoilers may be a high of 200%)

[Dark Hawk]

1st to 4th quetion who is that cat thing, what does it do, what is it's improtance, and why can only Tussake understand it.
5th question why can't Tussake can't log out. (I can't spell the names nor do I know that cat like one one)

[YashaOokami]

That cat Thing DOES have a name,and it's name is Maha.
Maha is a servant to Morgana,the voice that speaks to Tsukasa in the series.Only Tsukasa can hear Maha,no one else,it does speak,but not out loud where everyone can hear it,just Tsukasa.It's importance..uh to watch over Tsukasa by the order of Morgana.And why can't Tsukasa log out? Watch the series or check the spoilers section,that is something you can find on your own.Hope that helps somewhat
-bows and leaves-

[Dark Hawk]

Sí hace. Me ayuda tanto. Gracias. (translation: Yes it does. It helps me very much. Thank you.)

[YashaOokami]

bienvenido