just checked, befor "Nennformgruppen" (spazieren zu gehen, schlafen zu legen, fern zu schaun,...) >can< a comma be set, but is since the grammar reform not mandatory anymore and only used if you wanna emphasice the pause in the sentence. reason for that is, that a comma originally was intended to separate different information in a long sentence. hope this helps and is kind of correct as well. :)
The subject isn't spazieren... the subject is du. Some things in languages just aren't logical in another language and don't translate well. This something that will always be an issue when learning languages. Some things don't have a clear reason or explanation and just need to be learned; I think this is one of them.
My problem is with the usage of the word gehen. I believe that native speakers would use the word machen here. The use of gehen seems very English to me. In the Foreign Service language course, this question was asked this way:
"Machen Sie mit mir noch einen Spaziergang zur Universitaet?"
Translated as "Are you going to take a walk to the university with me?"
As a native I'd say that it's more used with "machen" when you already have a clear destination (like in that case the "Universität") but "spazieren gehen" is often used when going for a walk just for the heck of it, similar to "going around the block". For example "Ich brauche frische Luft, wollen wir eine Runde spazieren gehen?"