You might want to try to isolate where the problem is by restoring your device to new as follows:
1. Back up your device to iCloud or iTunes
2. Go to Settings>General>Reset>Erase All Content & Settings
3. Set up your phone as new and do not restore from the backup, to see if the issues you are experiencing are gone
If they are gone, then you want to "manually" rebuild your device by downloading your Apps and Music in small batches to see if you can isolate where the problem is (you can sign into iCloud to to sync your Contacts, Calendars, Notes, etc. but even when you do that, you want to check both before and after the sync to see if the device performs any differently once the iCloud data has been sync'd)
If they are not gone, then you probably want to take your device to the Apple Store to have them take a look at it, but in the mean time, you can follow steps 2 & 3 again but this time around set it up using the backup you made.
The iOS 9 update should not be affecting your 5s negatively. It is a smaller, more compact version and should be working better, not worse than iOS 8 did.
I also have an iPhone 5s, and I'm not seeing any issues at all with it.
Best of luck,
GB