For what you could do to attract birds without a feeder, I would suggest a birdbath, and we have a few questions on them already.
Yes, the concern of it being spread by rodents is valid.
Density of infected nymphs—the principal determinant of Lyme disease risk—varied significantly from year to year, fueled mostly by large fluctuations in total nymph density, which in turn depended mostly on fluctuations in abundance of acorns, mice, and chipmunks. Interestingly, though chipmunk densities are generally lower than mice, their numbers were the best predictor of total nymph density in the subsequent year, likely reflecting their inferior grooming skills. Overall, the results found that acorns were the best predictor of Lyme disease risk—stemming from their crucial role in supporting white-footed mice, chipmunks, and likely other small animals, which in turn provide large reservoirs for B. burgdorferi.
A New View on Lyme Disease: Rodents Hold the Key to Annual Risk
Peromyscus spp. (white-footed mouse) in the northeastern and mid-western USA, and other rodents (tree squirrel) in the western USA
Diseases indirectly transmitted by rodents