Was the shell or have the recent shells been soft or thin? If not, and since she passed the shell, I probably would not. But if they were, it would not hurt.
Do you have any pictures of the lash egg? Lash eggs are a product of salpingitis or inflammation of the oviduct. Here is some reading about it:
https://the-chicken-chick.com/salpingitis-lash-eggs-in-backyard/
https://www.bhwt.org.uk/hen-health/health-problems/lash-eggs-salpingitis/
That is not terramycin but a knock off that contains no oxytetracycline. There are several of those brands of ointment being sold to sound like terramycin/oxytetracycline.
Yes, zoologix and RAL labs (vet dna) can do pcr testing for many diseases when they send you materials to collect swabs...
Vets can’t help much if they brought a respiratory disease. Symptoms can sometimes help to identify if it is serious or not. Sneezing every few minutes for several weeks can be a sign of infectious bronchitis virus which runs it’s course over time and has no treatment. If you notice any eyelid...
Give her human calcium or Tums to help her expel the rest if the egg. Has she passed any of the shell and contents? I would gently feel for any shell pieces and try to remove them. You can lubricate her vent if needed. Clean her up with warm soap and water. Any pictures may help.
Can you give her a human calcium citrate tablet with vitamin d3 300-600 mg orally to see if it helps her to lay? Tums can be substituted. Has she been laying regularly? How old is she? Have you seen any lash eggs or thin shelled or shell-less eggs recently? Is her tail position up or down? What...
You can use a drop or two of hydrogen peroxide in the ear canal and QTips to try and clean it out. Plain triple antibiotic ointment or a bit of baby oil or mineral oil can be applied to the ear canal to smother any ear mites. Check it often to see if that needs to be repeated.
What are you feeding them including any treats? Could some be bullied from eating enough? You may want to change them to flock raiser or all flock feed for the time being, and place some crushed oyster shell out in a separate pan for when they need to start laying. Some scrambled egg bits for a...
I can’t see a prolapse. What are you feeding her? She might have some reproductive disorder that is causing pain or irritation. It is hard to know, but she sure seems to be mutilating herself and getting hurt by the other hen. You might want to treat her like the one in this article...