Can I get into my iMac without mouse?

You can login with just the keyboard. Use arrow keys to get the user, press Enter (or Return), type the password, then Enter again.

Keyboard shortcuts can do most mouse functions.

https://support.apple.com/HT201236

https://support.apple.com/guide/safari/keyboard-and-other-shortcuts-cpsh003/mac

If you have the extended keyboard, you can also use Mouse Keys.

https://support.apple.com/kb/PH25642

You only need to charge Magic Mouse 2 for two or three minutes and that will give you enough charge to work for a day or more. Then when you've finished work you can put it on longer term charge.

Basically no 🙂