When I put eye drops, my eyes become even more red! Is this normal?

Depends. Eye drops can be used to control redness, but depending on the type of drop, they can initially cause redness. If you are treating red eyes with a vasoconstrictor (visine(r)), it can eventually cause more redness. Some medicated drops for glaucoma or infections can irritate the eyes. Please see an eye doctor to evaluate the drop you are using as well as the condition you are trying to treat.
Yes. The redness under these circumstances is part of the eye's normal response to any foreign body introduced. See above answer to question regarding burning of eyes when instilling eye drops.