Is MHC1 knockout sufficient to prevent transplant rejection?
No, MHCII mismatch can also lead to transplant rejection, though through different mechanisms. CD8+ Tcells (that recognize class I) may become activated and directly kill donor cells*, and some NK subtypes can see the lack of self MHCI and also induce cell death.

MHCII mismatch on antigen presenting cells (APCs) brought along in the tissue may activate CD4+ (helper) Tcells, which do not kill directly. However, they can secrete a lot of pro-inflammatory cytokines which will make a generally hostile environment, and can also activate b-cells to produce antibodies against minor histocompatibility loci. 

As to your other question, lacking MHCI entirely would be really rough on an animal in the wild, but under laboratory conditions where pathogens can be managed a bit more, it doesn't matter so much. It's the same reason we can keep mice that have gross immunological deficiencies alive, where humans with the same defects would be screwed.