Rise is not available on Genie+. It is an ILL. That's a separate purchase.
If you were only going to use it for Haunted Mansion, Pirates and Tower of Terror, I'd say buying Genie+ is a questionable purchase. Nevertheless, I recommend that you do buy it and use it for other rides too. If you are spending a full day in the parks, you should easily be able to get 6-12 rides with it.
You can book your first Genie+ at your second park when its return time is 2:00 or later. You do not need to wait until 2:00 or until after you enter the second park. Today, Tower of Terror had a return time of 2:00 at 11:27 AM. Slinky had a return time of 3:45 at 7:02 AM. If you want to ride Slinky (even though I realize it isn't your top priority), you should book it as your first Genie+ shortly after 7:00. If not, I'd book one of the more popular Genie+ attractions at MK (Jungle Cruise or Peter Pan) as your first Genie+. Some of the meet and greets (Ariel, Mickey or Cinderella) would not be bad choices either, if you are inclined to do those.
There are no single rider lines for Haunted Mansion, Pirates or Tower of Terror. Sometimes, for ToT they will ask for singles when you're at or very near the end of the line if they need to fill an empty seat, but you have to go through most of the line with everyone else first. They might do that for Pirates too, but I can't recall it happening. The way the line is for Haunted Mansion, that wouldn't work there. As it is, some of the doom buggies ride empty because they can't get people into the seats fast enough.
Thank you for your response on the park hopper genie+ question
With that I think I have a gameplan.
Midnight purchase genie plus.
At 7:00 am, I'll book my first LL for Tower of Terror for around 5:00 pm if available. Is that even possible? I have a feeling that the LL return time slots only become available bit by bit through out the day. That will make my gameplan even more difficult. But I guess I could snag it, and just keep modifying it for later times, right? Thats kind of a pain..
At 8:30am(when DHS opens) I'll try to purchase Rise of the Resistance as an ILL for late in the evening. I don't know if they offer a 6:00pm or 7:00pm slot, but that is what I am going to shoot for. It might have sold out before I even get a chance since I don't have early purchase(staying at universal property instead), but that is what I am aiming for.
*I figured I would book LL first for DHS since they will have longer lines than MK seeing the MK has more rides distributed amongst the park guest population than DHS does. And it will be much hotter by then(late afternoon and evening).
At 9:00 am, I'll enter MK, but I guess I can't book another LL until 11:00 am, right? Two hours after the park opens, even though I booked one at 7:00 am. In the meantime I'll just stand in line at HM, and then maybe get a dole whip to cool down. Can I eat that Dolewhip whilst in line for Pirates?
At 11:00 I can finally book another LL. And then repeat every two hours or when I use the previous LL. Hopefully I can get Thunder Railroad. The rest I'll just fill in with shorter line stuff, People Mover, Carousel of Progress, hopefully see the Dapper Dans on mainstream,
I don't know how long it takes to get from MK to Epcot via monorail, and then from there to DHS via Skyliner, but I am just going to be safe and say an hour. I hope to stop by the Rose&Crown for a pint before taking the skyliner, and maybe a dessert from Fance. So if I leave MK via rail by 2:00-3:30, that should give me just a bit of time for a drink at Epcot, and maybe a dessert. And of course I can always try to modify the time for the TOT for a later slot if available.