for super high end LED screens that you'd see at like a festival, there is usually a dedicated VJ for the whole festival, who takes care of the syncing of music with visuals after getting feedback from the performers during soundcheck and whatnot. there are a slew of software for this that are used professionally, like VDMX and Resolume Avenue. if artists have their own specific set of visual art that is unique to their performance, they typically have their own VJ on payroll or a very, very meticulously made set of instructions or project files for the VJ software they prefer. again, they'll most likely go over the syncing in soundcheck and the pre-show set up.
as for how those visuals are made, i know a lot of graphic artists use Cinema4D. for what you want to do, that might be overkill. you could probably get by with some Adobe products to take care of your logo and superimposing it over whatever visuals you want, if you have any. you'll want to check with the venue you'd be performing at to see what their VJ situation is like, because I'm guessing for most small clubs they have a dedicated guy who does all that who you could supply your image/video to and they'd make it happen from there. i've heard that some artists carry those images/videos on a USB stick or other portable data storage device, in multiple file formats just in case.
