No. Not all verbs that use prepositions are separable in the sense that some forms are 1 word and other forms are separated into 2 words. There are no forms of this verb where houden and van are in 1 word.
Do you mean that in some cases the same verb can have two different meanings, one separable and the other non-separable?
"Van" is necessary to give "houden" the 'love/like' meaning. "Houden" alone means "to keep"
I am very fond of you. Ik houd van you. I am fond of cooking. Ik houd van koken.
Basically, it means I like or I love, but you say I am fond of :-)
So are the plural forms of verbs 'houden van', 'koken', 'lopen', also the infinitive forms of the verb?
Because you are asking to yourself if you like the act of cooking, it doesn't fit the conjugation of this verb in first person (you are not doing the action).
So if it is a continuous verb or infinitive, it ends with -en, like ik houd van lopen, or is it ik houd van loop?
I don't understand this verb at all, and I keep failing this lesson. Can someone please help?