Yes, you could view the batter life via the software.
Yes, you can remove the battery at your own risk and responsibility.
The warranty only covers defects of products caused by manufacturing. It does not cover regular wear and tear, including but not limited to, accidental damage, breakage due to abuse, improper usage, water damage, etc. The Warranty also does not cover any cosmetic packaging damage that may be caused through shipping of goods to the customer.
Please do not attempt to remove the battery on the wireless mouse unless you are okay with using it in wired mode only indefinitely. The warranty only covers factory defects, any accidental damage to the battery connector or mouse while taking the battery off are not covered by our warranty.
If you are okay with the warning and risk, here is a link to the video demonstration of how to properly take the battery off:
Please do not use any tooling and blunt force on it. pull the wire straight up with firm upward strength, if the wire on your unit is too short to firmly grip with your bare fingers, use a cloth to help. It's very important to use only straight upward force with no tools, or the side of the connector might break.