It is fine that way.

Yeah it is completely correct.

"I think I"m going to...." and "I think I will..." have the same meaning. The "I think..." part means that the speaker is not 100% sure she will do it. If she were less sure, she could say "I think I might..." or "I think I may..." In some contexts the "I think..." part implies a great deal of certainty (almost 100%). For example I can say: "I think she is an idiot for going out with him." Here I mean that I really believe she is an idiot, and I don't have any doubt. In other words "I think she is an idiot..." = "She is an idiot..." in this context. In this context "I think..." means "In my mind right now, I am having the thought that...."

You can say I will spend the rest of the day here I'm gonna(going to) spend the rest of the day here I want to... ....

You can say it that way for sure! You can also say "I might spend the rest of the day here". To say "I think I'm going" does allow room for someone to change their mind, whereas simply saying "I'm going to" implies a definite answer.