You need to be careful. There are many rules issued by the government in protecting trees. It is one thing if the tree is dead, but you might be fined for chopping down a perfectly healthy one.
Depends on the forest. If it is your own private property, it is probably legal (subject to possible restrictions preserving certain species or something like that). If it is public lands, such as a state park, the cutting down a tree is probably VERY illegal.
If you don't have permission then yes its illegal.
It is certainly illegal because you r destroying the most precious things and that area does not belong to you.