What would happen if a Russian TU-95 had an engine failure or other emergency close to the United States?
Losing an engine on a 4-engined aircraft is not overly dramatic. In fact the P3 Orion actually intentionally shuts off one engine in flight to conserve fuel- it's more efficient to run three of them at higher power setting.

I doubt that they would put their aircraft in a position that they could not withstand an engine failure. From Kamchatka to California is it around 6000km max one-way; The TU-95 has a range of 15,000km.

As for potential political mess, you can see what happened when a Chinese J-11 Fighter collided mid-air with a US Navy P3 Orion back in 2001 in what is known as the Hainan Island Incident.

A British Airways 747 lost one engine out of Los Angeles a few years ago, and almost made it all the way to London, but diverted to Manchester (although that event did cause a lot of controversy).