Yes. Only the female mosquito bites. The male feeds on plant juices to be used to fertilize the eggs of the female, which must then have a blood meal before she can produce viable eggs.
Some species of mosquitoes only bite birds, some cold-blooded animals and some prefer rodents, etc. Not all are capable of transmitting disease. Some are capable of transmitting several viruses and some are only capable of transmitting one virus. The virus must undergo certain metabolic actions within the mosquito before it can then be transmitted via the bite. Some viruses are not capable of undergoing this and some are not present in sufficient quantity in the mosquito to be transmitted. Some mosquito species don't bite at all.