"It seems that we will stay here tonight."
Translation:Parece que nos quedaremos aquí esta noche.
"Quedaremos" is the future tense of "Quedar".
The "nos" is a reflexive pronoun that is required in this sentence. The reflexive pronoun changes the meaning of the verb to show the verb is applying to the subject of the sentence.
A rule of thumb that ususally works is to think of the pronouns this way:
me - myself
te - yourself (informal)
se - himself/herself/yourself (formal)
nos - ourselves
So, in this sentence you could think of it as "We will leave ourselves here tonight." However, normally that is not how we speak so we would change it to "We will stay here tonight"
As I explained above it is not a reflexive verb, but pronominal verb. Here me te se nos os... do not mean myself yourself etc We consider a reflexive verb lavarse, peinarse, when the subject is the same as the object Wash yourself etc. In this case it is a different meaning. Quedar con alguien means to meet somebody quedarse en un sitio means to stay in a place.
1) A lot of references explicitly state "quedarse" to be a reflexive verb.
2) Reflexive verbs are actually the most common category of pronominal verbs. So, a verb can be both pronominal and reflexive.
3) While I am sure the examples you gave are correct (there are a lot of idiomatic uses for quedar), the primary definition of "quedar" is "to be left", and the primary definition of "quedarse" is "to remain" So the rule of thumb above is still valid, and is valid for most instances where we see these pronouns.
Yes, "quedar" is different from "quedarse". This is not called reflexive verb in Spanish but pronominal verb (reflexive is lavarse, peinarse... = when the subject is the same as the object) In pronominal verbs we just change the meaning of the verb when it goes with the pronoun. Quedar con alguien is to meet somebody, quedarse en un sitio is to stay in a place. I hope you understand my English. :)