[Rusty]

The sentences in question are using the 'ir + a + infinitivo (el infinitivo del verbo principal)' structure and the pronoun 'se' is tied to the verbo principal.
So, the question is whether quedarse, in the first sentence, is more appropriate than quedar, and, in the second sentence, whether seguirse is more appropriate than seguir.

quedar = remain, be left with, meet up
quedarse = stay (in one place)
How long are you staying in Rome? The futuro próximo form must be either 'te vas a quedar' or 'vas a quedarte' (the pronoun can be placed before the conjugated form of ir or it can remain suffixed to the infinitive). Without the pronoun, the sentence doesn't make sense.

seguir = continue (in this case)
seguirse = follow (as in infer (it follows that))
How long are you going to stay mad at me? The futuro próximo form must be 'vas a seguir'. It doesn't make sense to use the pronominal form of the verb.

Update: I see that Alec also provided the same answer.