You can run any laptop with, or without the battery. If you plan to use it plugged in most of the time, you can take out the battery and not worry about it charging & discharging all the time.
I would charge it fully the first time , remove it and store in a safe place, in case you want to use it in the future. Then use plugged in just as you are now.

As far as newer being faster, yes probably, but some slowness on your system now could be from Windows OS getting bogged down.. do you defrag the hard drive yearly?
That get s the files back in the best order for the system..

Usually the speed of the computer matters more for gamers or people having multiple programs open or a heavy graphics work..
for mainly online use the computer speed would not be a big factor. Then the wifi speed is more important, if you get a lot of buffering or not when watching videos/shows...