1. Je préfère cuisiner avec de l’huile. I prefer cooking with oil.

Here de l’ is a partitive article. It may be conveyed by ‘some’ (or ‘any’). Nevertheless,’some’ is often omitted.

The partitive article refers to an unspecified quantity of food, liquid, or some other uncountable noun. English has no equivalent article – the partitive is usually translated by the adjectives “some” or “any,” or may be left out entirely.

2. No, the sentence is not correct.

3. In certain constructions, the partitive reverts to simply de (or its contraction d’). One such is expressions of quantity. Cf.

Edouard: Rien de plus facile! Tu mélanges de la farine et des oeufs. Tu ajoutes ensuite du lait, du sel et de l’huile. Tu verses cette pâte dans une poêle. Quand la pâte est cuite, tu garnis la crêpe avec du fromage râpé, du jambon, ou des cèpes. Mais n’oublie pas Tammy, avec des crêpes, on ne boit pas de vin. Il faut boire du cidre et porter une coiffe bretonne!

Edouard: No problem! Mix some flour and eggs. Add some milk, some salt and some oil. Pour this batter in a pan. When the batter is cooked, fill the crêpe with some grated cheese, some ham, or cepes (mushrooms). But don’t forget Tammy, with crêpes, you don’t drink wine. You have to drink cider and wear a Breton hat!

Source: https://www.laits.utexas.edu/tex/gr/det5.html#:~:text=Partitive%20articles%20are%20used%20both,that%20are%20indivisible%20or%20uncountable.

Dimitris answered very well questions 1 and 2. Here are some elements on question 3 (avec cinq huiles… vs. beaucoup d’huile).

3. Actually, the difference between the two sentences — (2) «Il cuisine avec cinq huiles différentes.» vs. (3) «Le chef ajoute beaucoup d’huile.» — has nothing to do with avec, as it is a preposition and does not change anything. You could very well use avec with beaucoup: «Il cuisine avec beaucoup d’huile».

   In both sentences, the reason why there is no article (la/l’) is because the article is replaced by another quantifier, another determiner: cinq in (2) and beaucoup de in (3). Both replace the article entirely:

   Il cuisine avec une huile particulière. → Il cuisine avec cinq huiles différentes.

   Le chef ajoute de l’ huile → Le chef ajoute un peu d’/beaucoup d’ huile.

   You might have seen grammar explanations where beaucoup de is presented as an adverb + an article (because, of course, beaucoup is often an adverb, as in «Je t’aime beaucoup»), but it is much clearer to consider beaucoup de as a determiner (an indefinite determiner to be precise).

   Note that, as a determiner, beaucoup de should never be followed by an article: you should never say beaucoup des, beaucoup du, beaucoup de la/l’. (And ignore the extremely far-fetched cases proposed by some mediocre grammar websites: they refer to a different, and almost inexistent, grammatical construction.)