Yes he can. His sharingan enables him to see the hand seals movement of the enemy, where then he do the same seals, thus copying the jutsu. However, during Naruto's training of Rasen Shuriken, he told Naruto that he can copy the Rasengan but only up to the level where the Rasengan is on the palm in its sphere form. He cannot copy beyond that despite the fact that Rasengan is not a Kekkei Genkai jutsu. What this means is that the Sharingan is not the reason he can copy other people's jutsu. Sharingan helped him copying jutsu by seeing the hand seals, but what enabled him to perform the jutsu is that he has the elemental affinity for the jutsu. So far, it has been shown that Kakashi is able to use 4 of the 5 basic elements (fire, water, earth, and lightning). His main affinity is lightning, though.
Also, based on the manga where ninjas not from Village of the Hidden Rock used earth jutsu to stop the bijuudama, it seems that up to a certain level, people can use jutsu of elements that they don't have the affinity for. But, considering that Rasengan was said to be an A rank jutsu (or S, not sure) it is possible that people's element affinity itself has its level. Meaning that Kakashi had the affinity for 4 of the 5 basic elements with lightning being the strongest considering his only original jutsu (Raikiri a.k.a Chidori) is of that element. This fits the fact of him being able to use Rasengan, an A rank jutsu, but not being able to use Rasen Shuriken, an S rank jutsu which requires elemental transformation.