OMG with "veux" vs "vais' and "le" vs "les" my greek ear always ignores the difference and jumps to whatever comes first in my mind!
"I am staying here" should also be considered correct
OK, but It shows some extra determination. Probably in french it would be " Eh bien, je vais rester ici"
I agree - in english, the present tense is often used to imply a future action. eg 'I am staying at home tomorrow'
Present continuous, to be more precise.
Unless a timetable is involved.