Some possibly dumb Q questions
There are usually no dumb questions.
1. Since it's "live view", any filters will effect the light getting to the sensor, so you'd see the effect on the LCD.
2. All of the in-camera filter effects display on the LCD in LV -- pretty cool!!
3. It's a pretty commonly available battery -- luckily -- and they can commonly be found for under $6 USD each on Amazon or Ebay. I bought 1 Pearstone from B&H and 3 from Wasabe with their 120V home / 12V car charger. I've had them about 1 year, and they all are working pretty much like the original.
I bought my 3rd and 4th batteries from an Amazon side store for something like $7 for the two, including shipping.
If you don't use the "Power Saver" function (far too aggressive, IMO), you will need at least 3 batteries for a day's shooting. When I had just 2 batteries, and used the Q with a Canon EOS telephoto with an adapter, with "Power Saver" off, both batteries were dead halfway through the day.