Hey fletch, welcome to the board!
Regarding grappling at home, you can shadow grapple just like people shadow box, you just string together your grappling techniques into the air the same way you would string together striking techniques into the air of shadow boxing.
The issue that most people have who are new is basically having a very small grappling "vocabulary" (understanding of the feel, flow, and foundational grappling movements), so it's like trying to write a speech in a language that you barely know how to get the most basic statements across.
My suggestion would be to work on your basic ground mobility (shrimping, bridge and roll, sit out, hip heist, rolling, breakfalls, etc...) first, and try to string things together and just get comfortable and familiar moving your body around on the ground.
Next I would practice any grappling skills sequences/techniques/flows/chains (lots of names for how you string together skills in grappling) that you learn in class, focusing on the important technique cues that your instructor(s) gives you and really try to visualize yourself doing them to an opponent. At first these might be single techniques done repeatedly, but eventually will evolve to a "combination" of techniques geared towards eventually "trapping" your opponent in a submission (or takedown, or Sweep, or guard pass, etc...).
Finally, once you gain enough experience you can just full on shadow grapple like you see boxers shadow box.
As far as the qualities I look for in the ideal student:
2) Character (not interested in teaching bullies how to be better bullies)
3) Heart (perseverance/noting giving up when things get tough)
4) Consistency/genuine love for the Martial Arts (people are only usually going to stick with something for the long enough to get really good at it if they enjoy doing it)
Talent is nice of course, as it makes my job easier (as such people tend to pick athletic pursuits like Martial Arts up more easily), but I'll take a less talented person who embodies the above qualities over a super talented arrogant bully with a chip on their shoulder (even if they show respect to and are friendly with me) any day.
Hope this helps. Good luck.