If I target my Android app for version x, will it have any problems if the user uses an Android version above x?
What will I have to watch out for?
Yes, it's should work.
Updates to the framework API are designed so that the new API remains compatible with earlier versions of the API. That is, most changes in the API are additive and introduce new or replacement functionality. As parts of the API are upgraded, the older replaced parts are deprecated but are not removed, so that existing applications can still use them. In a very small number of cases, parts of the API may be modified or removed, although typically such changes are only needed to ensure API robustness and application or system security. All other API parts from earlier revisions are carried forward without modification.
Take a look at this