That 1st one is most likely Hylocereus undatus (pitahaya/pitaya, dragon fruit). What you see is the way it grows (and that's why I got rid of one I had :) It grows big, spiny, with those hard aerial roots. It needs to have support and be tied to contain it. I would grow it again only if I had lots of space, or (even better) if I lived in warm climate, where it would get plenty of sun and heat it needs to produce fruit. You can look up how they grow them on 'pitaya farms', trained up on trellises. There are also photos available of plants grown in containers and kept quite neat - if they are real. I am not sure if I would invest space, time and effort in my zone. Just to have a big, very prickly plant...Obviously, it is your choice to keep it or not. If you get it to bloom, flowers are very nice. They are big and IIRC, nocturnal.

This plant is used a lot as a rootstock for exactly what you purchased - grafted plants. It is very vigorous and probably easy & inexpensive for that purpose. It grows easily from seeds if you purchase fruit in a grocery store.

The first cactus you bought didn't have a flower on top if you had what I think you had. It was a different cactus grafted on top of another. The top cactus died but the rootstock survived and now it is growing like it normally would grow. It is a climbing/scrambling cactus so the growth you're seeing is normal.

The second cactus is technically not a cactus. It is a Euphorbia. Not unusual for some Euphorbia to have seasonal leaves. I don't have much advice about cutting the plants. I've never tried to do a cutting as big as that but seems like you're already familiar with the basics of doing a cut.

Keep in mind that Euphorbia plants often have milky sap that you have to be careful about. Most people aren't bothered by it touching their skin but it can cause considerable irritation if it comes in contact with mucous membranes and it is dangerous to the eyes. I suggest wearing gloves when handling the plant so you can dispose of them when you're done.