Hi Val,
Welcome to the Madness that is Animated Holiday Lighting.
As far as which Pixel controller to use, all of your choices are good. With both of the Falcon controllers you will need a RPi or some other similar small computer to run the show. I am not totally familiar with the Kulp controller, but I would presume the BBB integrated into the controller will take the place of the RPi.
Your AC Lights are typically controlled through Renard or DMX protocols. There are a variety of different "Renard" controllers, but that is NOT a brand name, rather controllers that run on the Renard (or DMX) protocols. With this type of controller you plug your light strings into the Solid State Relays (SSR's) which can be integrated into the controller or remote units similar to the Smart Receivers for pixels. Controllers and SSR's are plugged into your wall for power and Yes the RPi can run these controllers.
XLights and Vixen are similar but different software programs for sequencing the lights, both are VERY good and powerful. You only need to use 1 of the programs, although I would recommend playing with them both as the work flows are a bit different and you may find one more intuitive than the other. I personally chose XLights and I have been extremely happy with the system and support I receive.
Good Luck and feel free to ask lots of questions. You will find there are many ways to accomplish what you hope to achieve and there will be many opinions on the best way to accomplish them. That is part of the reason why new folks can get so confused when reading the forums.
Remember, HAVE FUN.
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