If a current modern day fighter were to have a total engine failure over a suitable landing site or airport, can the pilot safely perform a dead stick landing? If so, are they routinely given training for the procedure?
Here's an F-16 doing it.
Apparently there is an emergency APU that runs on hydrazine to power the hydraulics instead of a RAT. They mention it at the end because they have to call out fire trucks because of the hydrazine hazard.
I used to have an F-104 flight manual, which I must have loaned out to someone and never saw again. It could actually glide fairly well, with an L/D of about 4 or 5 to one at about 190 kt if I recall. It had a section on gliding engine out.
Yes, in fact it has happened successfully at least once I can think of. In 1997 an air national guard pilot landed an engine out F-16 on the numbers. It's not the most advanced fighter out there but it shares almost all the characteristics of the newest as it's reliant on electrical power and computers to fly.
As for training, fighter pilots are routinely taught engine out procedures at least to the point of restart, I do not know if they teach dead stick landings though, although I suspect not.
There's a youtube video of the approach and landing here.