1. Did you finally move into your new apartment?
2. Did you finally move in your new apartment?
(Can we use both 'in' and 'into' in the sentence? Which one is common?)
3. What about dinner at/in my place?
4. What about having dinner at/in my place?
(Are both okay? Which prepositions do we have to use? Do we have to put 'having' before dinner?)
1 and 2 -- Both are fine; "into" is the full meaning, but people will understand "in" as well.
3 and 4 -- We usually use "at" for this expression. You can use "having" or omit it, and the meaning will be the same.