Hyaluronic Acid

Yes, and it's often the ideal hydrator for oily skin because it adds water without oil. Oily skin is frequently dehydrated at the same time.

Both is fine. It's one of the few ingredients with no timing restriction and no interaction problems with other actives.

Serum if you want a hydration step you then seal; cream if you want one product doing both. If you buy the serum, you do need something over it.

Almost certainly applied to dry skin without a moisturiser on top, in air conditioning. Change those two things before concluding it doesn't suit you.