In a sentence like this:
Ich freue mich, mein Name fängt mit einem G an.
I want to add "dass", so that "fängt" should be moved to the end of the sentence. Will it be anfängt? So we shouldn't split the verb when it's placed at the end of a sentence?
The correct sentence would be:
Ich freue mich, dass mein Name mit einem G anfängt.
So you are right, it is anfängt. The verb isn't split.
Yes it will be anfängt.
As for the splitting... well it is not split, but you could argue that this is just for the lack of stuff in between.
Let's take an example:
Der Film fängt um 8 an.
Now in "spoken" past:
Der Film hat um 8 angefangen.
Now with a zu-construction (the meaning is odd but the grammar is ...top notch)
Der Film hat um 8 anzufangen.
And now without anything.
Ich weiß, dass der Film anfängt.
An doesn't move one inch in all these phrases. It has been split and put to the original end. The rest however gets inflected and moved around and after that it ends up even behind an. An and fangen have some sort of magnetic force if you will and this force makes them join again (like in the dictionary). But the modifications are done to the fallen-part only, so it has been split too.
Of course this is not official linguistic talk, but I think it is a helpful way to look at it.
