Interesting questions. Yes, chickens will eat ticks. Will they groom each other and find them? Maybe. I would not worry about ticks. I occasionally pick up a tick when walking through the grass and weeds, though since I got chickens, that happens a lot less frequently. My broodies take their chicks in the weeds and all over. A tick that had sucked a little blood would be pretty easy to see on one before they feather out. I've never seen one on a chick or chicken. You can check regularly if you wish. It won't hurt and it may help you socialize them. But ticks are not anywhere on my worry list. Some people get chickens in part for tick control, though guineas are better.
Mosquitoes are a bit different. They come out at night when the chickens are asleep. They are so small they don't get eaten a lot, especially at night when the chickens can't see. They can transmit diseases if there are other diseased birds around. I personally don't worry about Avian flu. It does exist but its basically been built up by the media because of the fear that it can be transmitted to humans. The occurrence is so rare it does not even make my radar. There are others, such as fowl pox, that can be transmitted by mosquitoes if other diseased birds are around. If you are showing birds, fowl pox is a specific worry since it can disfigure them, especially their comb and wattles. Since mine free range and my coop is well ventilated, I can't do anything about mosquitoes anyway. They are a fact of life. And I really think the risk is pretty low.
Chickens will eat anything that doesn't eat them first. Mice and frogs are high on mine's favorites list. They will eat young snakes, even poisonous ones. They eat spiders, wasps, and bees. I'm not saying there is not some risk, but they very seldom get hurt and very, very seldom fatally hurt. Mine seem to know not to eat certain things. They won't touch blister beetles, for example.
Life is not without risk. Mine have so much fun chasing grasshoppers that I am not going to concern myself with the off chance that something bad might happen with something they find. And so far nothing bad has happened.