I cannot do any form of push up (even wall pushes) if my arms are at my sides. The tendons behind my elbows “ping” and I have a sharp pain, but my triceps and other muscles don’t feel challenged. However, I can do good form full push-ups …. only if my hands are placed 10-12 inches away from my shoulders. I am worried about seriously damaging my elbows if I have a structural or injury issue.

if I’m reading your description right then it sounds like you’re maybe having some issues with your triceps
here’s an informative video
If you feel like that’s the case, I’d definitely aim to work on that specifically to alleviate those symptoms
For pushups obviously, as you wouldn’t wanna be flaring your arms out too much as you do them, but also for many other pushing movements where you might need them
And if even wall pushups are too much strain, you could potentially use a light weight (can be a water bottle etc.) or a resistance band to help you out (this can also be in the form of isometric holds)
Otherwise working up to being able to tuck your elbows more and more without pain with something like wall pushups could also help you alleviate the issue (but then again, you’d have to be doing the pushups relatively flared for a good bit which isn’t the best)
I hope this helps you out - in case it doesn’t or in case you wanna be in on the safer side, I recommend seeing a physical therapist to help you evaluate the situation better and get you a specified exercise plan for it :)

Stretch your biceps mate

Your triceps are underdeveloped. They are a synergist muscle in the pushup exercise. Id hold dips for time to strengthen the tendons and ligaments in your elbow and then docus on lowering the angle for your dips over time. This will allevaite the imbalance
