When the virus integrates its DNA with the hosts and enters the lytic pathway, do the viral proteins that produced destroy the cells DNA? Do they deactivate it? Also does the cell function in the same way as before?
A virus does not typically destroy its host's DNA. Rather, the virus makes the host cell express viral proteins at an elevated rate compared to the host's own proteins. This is usually achieved because the viral genome contains a strong constitutive promoter, possibly coupled to an enhancer sequence. The virus may also preferentially insert itself into transcriptionally active sections of the host DNA.
As I said earlier, expression of the host's own genes as a whole is not directly regulated by the virus, though specific genes may be up- or down-regulated (immune-system related genes are often primary targets). However, the host's genes will usually be expressed at a reduced level because of the resources being diverted to make virions.
Remember however, that the virus relies totally on the host's machinery to make viral proteins. So, anything that kills the cell before sufficient viral particles are assembled is going to be selected against. This doesn't mean that the cell is healthy though. A significant chunk of the cell's resources have been hijacked by the virus, and a virus infected cell is typically just limping along, clinging to life, as it were.
Depending on the virus, once sufficient virions have been assembled inside the cell, the virus may enter a lytic phase, where the cell literally bursts to release the virions into the surrounding environment to infect new cells. At this point, the cell is obviously dead.