Answer: The short answer to the first part is: no.

The longer answer goes something like this: a piece is said to be in a certain key if it creates the sense that the tonic harmony of that key is "home base." That will be the feeling you get if the music repeatedly "cadences" in that key. The kind of cadence I mean is where the harmony moves from the dominant (major chord built on the 5th note of the scale) to the tonic (the chord built on the tonic note of the key). Especially, of course, in the final cadence at the end.

But in between the beginning and the end all sorts of things might happen, and those could involve borrowing notes that are not actually in the scale belonging to that key. In popular music one of the most common borrowings will come from using a "secondary dominant," also known as "V of V." For example, in C major you might strengthen a change from d minor (the ii chord in C) to G (the dominant in C) by sharping the F in the d minor chord so that becomes major and sounds like it could be the V if you momentarily imagined yourself to be in the key of G. If you kept that up and repeatedly cadenced to G then you'd have "modulated" to the key of G. But if you just do this for the moment, to add some fun to that D chord, you're borrowed a note, F#, from the key of G and yet you're still in C.

Still using C major as the example, other common borrowings might be G#, to make an E major or E7 chord (V or V7 of vi), and C# as part of A or A7 (V or V7 of ii).

A little less usual would be temporary borrowings like Ab as part of a vii 7 dim chord (B,D,F,Ab) that moves to I much like the V chord does. And there are more, generally described as "altered chords."

What all those borrowings have in common is that they are just momentary effects that don't take you far enough away to be called a modulation. If you do use the same altered note repeatedly, for example using the chord pair "V of V, V" several times in a row, then you might give the impression that you're setting up camp in a new key now, and that would be a modulation to the new key.

Answers like this one just seem to go on and on. Not sure when to stop.