Don't know if this will help you, but I use "vários(as)" when I mean "several" and "muitos(as)" when I mean "many". I would have translated that sentence as "I have several questions." If I wanted to say "I have many questions.", I would use "Tenho muitas perguntas." as you wrote below.
It's quite confusing, because (according to WorldReference). In Portuguese, "many" is indeed one of the meanings of "vário(s)/ vária(s)". In other languages though, "several" really means "some, but not many". While "various" means "some different ones, but not many". Therefore our confusion at first. We have to accept that the Portuguese native speaker (not just the Brasilian one), doesn't really diffrentiate here. So WE can make the difference and make it easier on us while speaking, but have to keep an open mind, that when a native speaker says "vários" , he or she may mean "many"...