Running a laptop on AC only is fine, is if you plan to do this more than on batteries, it is advisable to remove the battery to save wear and tear, put he battery in once a month to top it off to about 90%, but do not let it go below 20%.
I have an HP performance laptop that is 5 years old, still on the original battery that holds a 98% charge doing the above.
As for using a 3rd party, yes they may not last as long, but can be cost effective against an original manufacturers part.