Better to use FragmentActivity (or ActionBarActivity) or system Activity when targeting min API 14 (ICS)?

some support library classes like FragmentStatePagerAdapter is not available on the system framework, and it needs the support Fragment instead of the system Fragment.

That is incorrect. There are two versions of FragmentPagerAdapter and FragmentStatePagerAdapter. One set (v4) uses the backport, the other (v13) uses the native API Level 11 version of fragments.

I wonder if I should make my activities inherit from the system Activity

Yes, unless you need nested fragments. Those did not show up until API Level 17. If you need that capability, you will need to use the backport, until such time as your android:minSdkVersion is 17 or higher.

If your min api level is 14 you can extend Activity. You use Fragment from support library below api level 11 in which case you need to extend FragmentActivity.

If you want action bar below 11 then you need to extend ActionBarActivity