I'm playing D&D 5e homebrew adventure and I bought a riding horse and a cart. Can a horse that is pulling a cart use the Dash, Dodge and Disengage actions in battle? If not, are there any rules about it?
The best answer is, ask your GM.
Or even better, next time you play, try it. Let's say you've named the horse Bucephalus. Next battle, on your turn, say something like "Bucky and I dash in, while pulling the cart. I leap off, and attack the goblin on the drop. Bucky uses the rest of his move to get out of range." See what happens.
The best summary of DnD is on page 6 of The Player's Handbook, "1. The DM describes the environment. 2. The players describe what they want to do. 3. The DM narrates the results of the adventurers' actions." So try it.
Only, you know it's a riding horse, right? A riding horse, a draft horse, and a war horse are different critters. There's no reason to assume that a riding horse is trained for anything but riding. You can harness it to a cart, but pulling a cart isn't what it's trained to do. Your GM could very reasonably rule that your riding horse isn't very good at pulling the cart. He doesn't like it, maybe. Maybe he'll pull it, but he's skittish.
And a riding horse definitely isn't a war horse. In the real world, horses famously run away when spooked or injured. You see it in movies all the time. The horse gets spooked and runs away pulling the wagon, for instance. That might be what happens if you try to get Bucky to do anything anywhere near monsters or other scary things. First you strapped this big cart to him, then you try to get him to run into a bunch of orcs while dragging it.
Now maybe your horse doesn't happen to be an average ol' horse. Maybe it's particularly strong or smart or fearless. Maybe ol' Bucky is the DnD equivalent of Trigger or Silver. You could ask your GM, "is there anything special about Bucky? Can I roll?" Maybe you'll get 100 on 1 to 100, and then maybe ol' Bucky is darned near magical. Maybe he talks. That would definitely be a horse of a different color. Or maybe Bucky is just a horse, of course.
Finally, no matter what, encumbrance has definitely got to be an issue. Ol' Bucky dashes away with a cart and there's a good chance that the cart ends up in front of the horse, or maybe even on top of it.
Beyond encumbrance rules I can't see anything RAW.
As a GM, I'd be inclined to say that if the cart was laden then the horse wouldn't be able to dash from a standing start as it would need to get the cart moving.
I would also be disinclined to allow dodge since I think the fact that it's attached to the cart in some manner means its ability to move around is hindered. Either that, or the dex save would just be a regular one rather than at advantage.