My questions are:
Can the media center, accessing directly the NAS shares cause any integrity problems?
No. A NAS should be able to deal with that.
(A SAN which an directly exported block device and the wrong filesystem on it might not, but a NAS operates at a different level and should be fine)
If I buy a 5 bay NAS, can I use 4 disks with RAID 6 for storage and 1 disk individually to run some services, like web server, FTP, ...?
You can.
Would it be a good idea?
No. First because using a four drive RAID 6RAID 6 array is wasteful. You only get the storage space worth of two disks from four drives. RAID 10 is also quite safe, offers the almost the same protection (1 or 2 disk failure, depending on which disks fail) and is a lot faster.
Secondly because you do want an external [off-line] backup.
Even two spare disk in the NAS are not going to rescue you data if there is a fire, if lightning hits or if a thief steals your equipment. The only truly safe solution is to have a second backup at a different location.
Thus I would not use 4 disk in RAID 6. Either use 3 disk in RAID 5, or 4 disk in RAID 5 to get more storage space, or 4 disk in RAID 10. In all cases make an external backup (e.g. a simple yearly copy to a disk which you then store somewhere else. Or a cloud based solution where you rsync the changed data to a different place.)