Isabel- I believe your host plant is a type of "cereus" cactus. You can take care of it the same way as most cacti plants. Place in full sun, provide a Soil or medium with excellent drainage, water appropriately. Depending on your area your growing needs may be different from what I do so, I don't want to be too specific. I water most of my cacti once a month sometimes longer than that growing it outdoors in a container in San Diego. Your situation may be very different so, the more detail you could provide would be helpful.

On your 2nd picture the cactus is growing what is known as "aerial roots." It's a normal occurrence with a lot of cacti and other succulents. They will eventually dry and fall off on their own if that's what you choose to have them do. You can detach the piece and plant it if you wish making it a new plant. It's a personal choice with no right or wrong to it.

if you detach it, I would wait for at least a week or 2 before planting it. The area where it's detached from needs to "heal off" as they say or form a callous first.

The 3rd plant in that awesome little concrete planter is called Eurphoriba tirucalli. It has 2 popular names: 1) pencil cactus, 2) Firesticks. It usually turns orange and red when grown in full sun. Be careful if a piece is broken as the sap is toxic. Do not get it anywhere near your face.