Yes. Particularly viral infections are very contagious (classic pink eye with the adeno virus for ekc), thus keye components: good hygiene, change towels, bedding, etc.
Certainly is! Very possible to transmit by objects or touch. For example "epidemic keratoconjunctivitis" is caused by adenovirus and has earned that well-deserved moniker. Wash your hands - you don't want to be known as the "typhoid mary! ".