2. Er muss nächsten Sommer umziehen.
This is the Best option. Gramatically you have to use "Modalverb" first, and then Verb in "no-person" form is in the end of your sentence.
Senstence 1 would be correct if it weren't incorrect - it should be "Er zieht nächsten Sommer um". This Verb is um|ziehen, and in sentence it is: ich ziehe .... um.