Do the amount of coins I posses go up if I stay. - It will not depend if you stay. It's depend on how volume will build up again on the coins you are holding.
Do I have to wait for the coins to go back to their original price that I bought in order to make a profit? - No. There are so called correction price. Sometimes waiting for target price can ruined your trading plan and strategy. Take the risk.
What are the reasons of holding if prices are down? - Well you don't want to sold your coins on much lowered price than you bought it.
Patience is the key. Altcoin price come and go but there are alts that can be considered as good for long term holding.