Yes you will need the filter, hopefully it is a efficient one. Be sure to keep it clean, it's probably the paper filter variety. The get clogged fairly quickly. Keep all the junk and vegetation that you can. All that stuff will eat up your chlorine. If a test kit is available get one to monitor the water quality. questions just ask.
Is the solar heater effective?
hopefully the solar pump is a by-pass pump and how efficient it will warm the water depends on how much surface area you have on the panels.
Should I buy a ground mat for the pool?
I'd get on only if it protects from puncturing the pool, it's a good idea prevent rocks and stuff from weakening the bottom.
FYI I was Chief of the Recreational Waters Program in Los Angeles County, California for 15 years responsible for inspection of 18,000 commercial swimming pools.