No question is a silly question!
:idea: Let me first answer your last question: Can the *te-form* be used alone?
Answer is, YES!
You can use the *te-form* on its own for __informal commands__. That is, you use it when you want to ask an action to be done. Such as, *Tabete* _("Eat!")_ or *Mite* _("Look!")_ Of course, like in English, depending on the tone of your voice the request can be an urgent plea, or a gentle command.
:idea: Now, let's talk about the phrase: *Kite iru no desu ka?*
The *te-form* of a verb plus *IRU* makes a two-verb combination that generally expresses an action in progress - equivalent to the English _"am ~ing/is ~ing/are ~ing"_ such as if you want to say, _"I am eating"_ you'd say *Tabete-iru* or _"Taro is sleeping"_- *Taro wa nete-iru.*
Another use of the *-te form + iru* combination is with momentary verbs - such as *kuru* _"[someone] comes"_ or *aku* _"[something] opens"_. The helping verb *IRU* in this case implies that the result of the te-form action __"exists"__. The English equivalent may translate as _"has/have ~."_ Sometimes, it may also translate as an ordinary present or past tense - _"is/are ~ ."_
So, *Kite iru* is literally _"has come,"_ but it's usually thought of as _"is/are here"_
*O yasumi de kite iru no desu ka?* thus means _"Are you here on vacation?"_