I'm not a novice shooter but previous precision shooting experience didn't emphasize a hard hold, loading bipod method of shooting. More sling stuff, or resting rifle on rucks, sandbags, or other shooting rests. Bipods were simple, non pivoting, nothing like the current stuff.  

Anyway, I have a 16" 308, relatively light weight precision rifle (about 10 lb loaded), so it will probably jump more than a normal rifle. But I've been unable to stop it from hopping, even when seriously loading the Atlas bipod. I've been using a very hard hold on the rifle, gripping the crap out of the pistol grip, pulling it really tight into my shoulder, and firmly placing my cheek on the stock. As far as I know, I'm directly behind the rifle.

My questions: Can excessive pressure on the buttstock from cheek weld cause the bipod to hop to the left (I'm right handed)?
How hard should I be gripping the rifle? How hard is a hard hold supposed to be? Death grip or just firm?
How relaxed are the shoulders and traps supposed to be?

ETA
Fixed, turns out I was pressing my cheek too hard against the stock. I built up the cheek pad a bit, and applied less pressure to it, and pulled straight back like normal, and was spotting shots at 100 yards with scope at 18x.

So yes, excessive cheek pressure can cause bipod hop.

Start here, these are both relevant to your questions.  In short, precision rifle shooting no death grip at all.





http://artoftherifleblog.com/natural-point-of-aim/2011/09/natural-point-of-aim.html





http://www.accuracy-tech.com/loading-a-bipod/



ETA  Here's a simple way to illustrate how extra pressure can mess with you.



Take your support arm and hold it vertical.  Now wrap you trigger hand around your support hand wrist.  Press you trigger finger as if you're firing the rifle. Feel the movement on your wrist and notice that the thump puts pressure on your wrist.



Now move your thumb to the same side of your wrist as your trigger finger not touching the wrist, now press your trigger finger as firing the rifle.



Notice the difference?  In long distance shooting that little additional pressure can account for differing points of impact.



 

Can excessive pressure on the buttstock from cheek weld cause the bipod to hop to the left (I'm right handed)?

If the rifle is hopping to the left, then you are not properly behind the rifle. Perhaps it is making contact with your shoulder improperly, perhaps you are not loading the bipod correctly, perhaps it is your forehand grip, perhaps it is your trigger grip, but something about you is making the rifle hop. You will know what your NPA is when the rifle quits hopping left or right. You will know what the proper grip is when the rifle quits hopping up or down.

How hard should I be gripping the rifle? How hard is a hard hold supposed to be? Death grip or just firm?

Most of the force (90%+) should be absorbed by your shoulder, with just enough grip pressures to allow the gun to land back on target after recoil.
The harder you grip the trigger finger, the less sensitivity to trigger release you will have, and the higher the probability that the trigger finger will push or pull the gun off line.

How relaxed are the shoulders and traps supposed to be?

Relaxed enough so you can hold them in those positions almost indefinitely, ready enough to absorb the recoil event(s) without knowing when they will occur.

You would be very wise to take some precision rifle classes to learn the basics. I would also recommend getting on youtube and watching shlowlights videos where he talks about all this stuff especially body position and recoil management. These videos will show you the basics of body position and trigger hand.

proper hand postion and trigger control

recoil managment
more recoil

I've watched the Snipers hide videos, didn't work even when I tried following the principals. I know I need someone qualified who can coach me one on one but I can't justify spending over $1k and taking vacation time to do it.  

When it comes to the fundamentals, other than this, I'm pretty good. I have a lot of "formal" training, am B4 qualified in the US Army, and shot on the local USMC shooting team when I was in the Corps. I understand perfectly about NPA and trigger control, I just can't seem to get it to work when shooting bipod supported prone. Magically if I shoot slung up or from an Alt position with shooting sticks, I don't have that problem. I just can't seem to get this bipod hop thing down. When I was in the Army, they really didn't focus all that much about recoil management, it was 90% fieldcraft and if you could put 3 out of 5 on a 1" pasty at 100 meters you were good to go and my unit was shitty in supporting us for ranges so we just didn't get to shoot enough or get quality marksmanship training.

I wish there was someone local that I could give $100 and have them spend an afternoon critiquing my shooting positions. I just need that little bit of training and I'll be good. But an expensive basic course at Rifles Only or something similar is just not realistic for me.

Rifles only is epic training but you might try to find someone local or close to you for a day or two class and you will spend a fraction of rifles only.

RO has a new prec. rifle clinic, only 1.5 days and $350. Maybe I'll do that next fall.

there probably is.  did you say where 'local' is?

In for "local"

For those in California I'm happy to go shootin' with someone and teach them how to do prone off a bipod or any other style of rifle shooting. My shooting areas are in central California (amazing full featured range) and Mojave (extreme long range, primitive conditions). You should be comfortable traveling to one of those places if you plan to ask me to give you a lesson. I can do a solid 1-day long range rifle clinic for up to 10 shooters. A 1:1 day of training is free. That's just homies out shooting which I'm always happy to do. If you want to bring a group then it's $150 each for 1-day and I provide additional instructors as needed and a BBQ tri-tip lunch. PM me if you want to discuss it at all.

For 1:1 stuff, I'm in NorCal. I'm sure a few hours drive isn't a big deal for me if I get to go shooting so the whole top half of the state is within range.

If anyone can point me toward a good instructor in the east Texas area, or Houston area, I wouldn't mind the same. Feel free to shoot me a PM if so.

I'm near College Station TX.

Keep an eye on the Tiger Valley curriculum.  He has an awesome range and does classes throughout the year.

 
I'm also searching for a legit Instructor for a crash course that doesn't cost me an arm and a leg, Texas-Fort Worth area. Also, I've been searching for a range that reach out to 400 yd or farther, not $400 annual membership.




Thanks!


 

Those were helpful. Having a bear of a time controlling recoil in a 7.25# .308.