is it open right now?

The birth home is open 24/7 for outside viewing but the tours are during the day and a maximum of 15 people per tour and you got to register at the visitor's center for the tour but get there early because the tour fills up fast.

But is it open for tours in this season? No goverment shutdown?

Yes it open from 9 to 5

Not sure of the hours there. I recall they closed around 3 est

How much does it cost to go inside his house

You gotta go beautiful place in history.

Can I visit tomorrow or do I need to register first?

No

You don't have to worry about government shutdown it's the 21st century and the opening hours is 9 a.m. to 5 p.m.