I would like to elaborate @Olafants answer a bit.
Question 1: gehören + direction (not necessarily accusative)
The construction in question is not necessarily gehören + accusative, but gehören + direction. In your sentence, the accusative is governed by the preposition an, not by gehören. The direction could as well be given by an adverb instead:
Auf und Abs gehören dorthin, nicht hierher.
Gehören + direction is an idiomatic phrase with several meanings and no complete equivalent in English. This insight is crucial here because in your example, two slightly different meanings are made use of:
- Auf und Abs gehören an die Börse states that ups and downs should happen on (and be limited to) the stock exchange. The DWDS gives this meaning as subitem of § 3 only by an example:
⟨jmd. gehört ins Bett⟩ jmd. müsste eigentlich ins Bett.
- [Auf und Abs gehören] nicht in den Kopf, however, matches the negation of the parent item in the DWDS, 'to be in the right place, to fit'. Thus it translates to 'to be out of place, to misfit'. The negation is actually so common that the Duden manages to give only negated examples for this meaning (§ 3) besides the variant mentioned above:
- das Fahrrad gehört nicht in die Wohnung
- das gehört nicht hierher
On the whole, I would translate your example sentence as:
Ups and downs are supposed to happen on the stock exchange, not in the head.
Question 2: Die Auf und Abs (shouldn't it be Auf-und-Abs?)
First of all: yes, it's plural, and seeing that it is a fixed phrase, the plural -s must also modify Auf, otherwise it would read *das Auf und die Abs. This aspect seems quite strange to me.
I found very little resources on the orthographical and grammatical properties of pluralized phrases that consist out of multiple nominalized adverbs. The Duden even introduces its online article on that topic with Doch müssen bei solch einem Hin und Her nicht Bindestriche her? without ever picking up that question again. Apparently, the Duden-Grammatik doesn't concern itself with the plural of multi-part nominalized phrases either. While das Auf und Ab is not hyphenated for some reason, to me it seems necessary to do so if the composition is pluralized as a whole:
?die Aufs und Abs
The latter parallels ups and downs in English, but virtually never occurs in German.