Can I rename the groups in the category sheet by typing over them without or is there another way? Also, when I make this change will this make the change in all of the other sheets that have the same group names? Will the tags that are attached, transport over also?
You should be able to change group names in your Categories sheet and see those changes immediately reflected in your Sheets templates. The one exception may be reports generated by the Tiller Community Solutions Add-on if you have those since the data is static.