1. I've finished all the v11 content with Alice and Kate is only at 3, so maybe a bug...? Like a lot of stats in the game that were implemented a long time ago, this is another one with no effect. I don't know if it's a bug, an error, or just an oversight, but nothing Kate does with Julia or Sophie or Alice seems to increase the interest in women stat or lewdness stat for me, and the scores don't seem to matter in terms of allowing or preventing Kate from taking any steps. It can be ignored for now, though I wonder if it will go up after the webcam show in v13?
2. Are you a paying patron? Alice's quest is v11 content and unless you pledge on Patreon, it's not available yet. If you are, sorry I can't help with this one. When I started v11 with the "rebooted" Alice-Lisa content, I was pleasantly surprised at how easily it unfolded. I don't recall if there was anything specific Kate needed to do other going to school and moving in the hallway a couple of times after school.
3. The communication stat goes up when you flirt with certain characters at school at the right time. I'm not aware of anything that this stat enables or prevents other than the flirting points you can get...?
I got the 2nd computer several versions ago and at that time there didn't seem to be any benefit. Maybe now or in the future Kate's scores will increase more each time she studies, or homework will take less time to complete, but for now there doesn't seem to be any benefit.
4. I'm not sure what you mean here... It's been a long time since I've played that quest, but this quest played out relatively smoothly as well, with Kate learning enough about the school to prompt Andrew to "recruit" her to get on everyone's good side to be a spy/influencer. (LOL I'm not saying this isn't a dumb plot. I just don't understand your question about Kate reading the files).
Anyway, hope that helps!