Why is van used here?
'Houden van' is the translation of 'to love'.
is the verb then vanhouden?
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.
So it is just an expression, like a phrasal verb in english?
Do you mean that in some cases the same verb can have two different meanings, one separable and the other non-separable?
what is the 'van' purpose?
"Van" is necessary to give "houden" the 'love/like' meaning.
"Houden" alone means "to keep"
why koken? is koken the sort of ing-form of kook?
'Koken' is the infinitive.
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?
Why is not possible to ask like.. Houd ik van kook?
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?
Is it love or like cooking?
Both are possible
I love cooking isn't possible?
It's a question. Do I love cooking?
Can I also say "Hou ik van koken?"
Yes you can, but houd ik is the more formal way. Hou is spoken language.