I also would not squeeze Windsor and Bath into the same day. There is plenty to do in Windsor for an entire day. Tour the castle in the morning, then walk into Eton to explore. The walk is easy and pretty, featuring a river and swans. Have lunch on the way: there are nice options in Eton. I would think Eton College might be of interest to kids of your kids’ ages. :-)
Yes, the train station is right across the street from the castle, so no need for a car.
(The station has been developed into what is basically a big shopping mall - might be fun to explore.)
There are large parking lots adjacent to the train station. I am not certain of what they cost, but if you do choose to drive (I wouldn’t) there is parking.
An incredibly cheap option to Windsor is the £2 bus - check out the LondonLine 701 and 702 bus online for more info. https://www.reading-buses.co.uk/services/RBUS/701
Given the size of your party, this would make the excursion more affordable. The bus takes about an hour each way if there’s no traffic.
For Bath, I’d suggest at least an overnight jaunt. There is a lot to see! I haven’t been in a few years, though, so will defer to others for more current and specific travel info.