Your plant is not a cactus , but an Euphorbia grandicornis. If you want to remove the offsets beware of the poisonous latex. Do not work with this plant without protecting gloves.
Eduart is spot on with the ID of what looks a lovely plant in excellent condition.
Eduart is quite right to warn you regarding the milky latex sap of all Euphorbias. It is extremely poisonous but forewarned is forearmed. Do not worry just take adequate precautions. Do not handle the plant if you have open cuts or wounds on the hands and do not rub the eyes after handling the plant.
It sometimes worries people when we talk about the poisonous sap, but do not worry you have obviously had the plant for some time without any trouble at all. The plant is not going to jump out of the pot and poison you, but it will certainly make you jump if you handle it without due care and attention.
Enjoy a beautiful plant and just treat it with respect.
Now that we have got that out of the way we will try to answer your questions:
It would seem from the look of the plant that you need very little if any advice on how to keep it, the plant looks in superb condition. Whatever you are doing is certainly working. Just a tip on watering: Allow the plant to dry out between waterings, before giving it another good soak and never allow it to stand in water (a saucer) for any length of time.I personally would reduce the watering dramatically during the winter. But if it is in a warm room give it a little about once a month.
As regards the "babies" this is the natural way the plant grows and the babies will eventually make quite large branches of the plant. If you wish to propagate the plant these can be removed with the sharpest knife you have, making a clean cut at the narrowest part where they attach to the stem. When you do this the milky latex sap will bleed from both cut surfaces. A good way to stop the "bleed" is to spray the area with tepid water. The small plantlets that you have taken off need to be left for about a week for the cut area to heal over (this is vital), they can then be set in a very sandy soil or a substance called Perlite, where hopefully they will root in about a three week to a month but make take a little longer.
The best time to do this would be late spring/early summer.
If there is anything you are not clear about please let me know.
