My recommendation is: be careful.
Replacement batteries always work best if they come from the original company that manufactured the laptop. Your replacement battery isn’t (as far as I can see), and thus might cause problems with your laptop. As long as everything seems to work mainly fine without any errors, I’d leave it in, but the second you start smelling smoke or hear strange sounds, get it out immediately.
Keeping the old battery would be a good idea if you could keep it in a cool and dry place, otherwise dispose of it responsibly.
The problems you are having with sleep and shut down might be related to the battery, but also might have to do with the upgrade to windows 10. If you have some time, first try putting your old battery in and then seeing if you have power issues with sleep and shutdown. Otherwise you could install a different OS (Such as Ubuntu) and see if the problem still occurs there.
All in all, you can never be too careful, but honestly I think you’ll be fine! The best option (also the priciest of course) is to order an original battery for the laptop.
Carpe Diem,
Minas