Problem:

T-cells recognize the MHC molecules and body's own peptides.

When it doesn't, it alarms the immune system. But do T-cells express MHC molecules ? If so, how are they using it?
If not, what happens when a virus infects T-cells? (Yes, I am confused about HIV infection mechanism too. They escape from immune response by altering their genes and disrupting the MHC-peptide bonding. I can see that this can work in macrophages. But what is the situation in T-cells ?)

Please explain it.