I have a Hensel Integra Pro Plus 500 Studio kit. This kit has 3 500 watt monolights. I hope your familiar with this set up.
1. From my best understanding, I would use 2 monolights for the green screen using umbrellas, soft boxes, barn doors or just plain 7" strobe reflectors? Which one is best?
2. How are your FXhome studio light kits compared to mine?
3. For my talent, I can only use one monolight and use a hand held reflector on the opposite side.
Your primary objective with lighting the screen itself is to have the light be as even as possible. If you were to use one light with the 7" reflector, you'd get a bright area where it was pointed ... then the screen would be progressively darker for pixels further away from the light's center on the screen. Not good.
You could likely do fine placing a single light above and slightly behind your subject, pointing at the screen with the soft box. This should cast a very even, shadowless light over most of the screen. These would leave you with two lights to light your subject, but you may see some light falloff towards the lower end of the screen though.
Lately, I've been shooting mostly using a strobe on either side of the subject, with softboxes or umbrellas, pointing in at the subject. The two lights tend to cancel out any shadow directly behind the subject .. though there can be shadow out on the edges of the screen, this allows for an easy setup with the subject's feet making contact with the screen on the floor.
Hope this helps.