This is what I imagine would happen if you magically found a way to alter all the DNA in an organism at once:
1. Rejection: Eventually the organism's immune system would produce new immune cells. These cells would come to recognize the original body tissue marker protiens as foreign, and you'd have the body rejecting itself.
2. If rejection was suppressed (say, through some serious drugs) you'd probably end up with all sorts of cancers. DNA has many epigenetic control proteins that control expression by physically existing on the DNA strand, blocking transcription.
3. If that wasn't enough, and you managed to get all the DNA function correctly, then you'd end up with cells that frequently repopulate themselves beginning to mimic that of the donor DNA. Specifically, skin cells, blood cells, etc. Cells that rarely reproduce, such as nerve cells, would have less dramatic changes. Assuming the two organisms' DNA is similar enough, there might not be any difference at all.
So, in your frog/mouse example, I'd say it's probable the mouse would eventually develop frog like skin and blood, but it's skeletal form would remain that of a mouse.
Just my guess anyway. It's all hypothetical anyway.