I had some concerns as I got a flu shot and am scheduled for my Pfizer covid booster in a couple of days. What they told me was that they did have at one time a recommendation to have 14 days between shots. But that now, you can get them closer without issue. They do not interact. However, they like to do them in separate arms. So, I'll have flu in left arm and they will give covid in my right. Because some develop pain and rash and they want to know which vaccine caused it.
Now, I DID find this. https://www.mayoclinic.org/drugs-supplements/dexamethasone-oral-route/precautions/drg-20075207 That is for the oral dexamethasone. But they don't want you to get any vaccines while on that unless a doctor approves it. Different than an injection but I would most certainly run it past your doctor before your vaccine to make sure.
This doesn't make any sense. If you mean lincoycin, that's a very strong antibiotic and doesn't treat viruses, it treats bacterial infections that are resistant to other antibiotics. You also wouldn't get a steroid for a cold. Neither of these would be used for the very mild common cold, you'd be told to go home, rest, and drink plenty of liquids. You might contact your doctor and see just exactly what you were treated for, as it might help answer your question. Steroids do suppress the immune system but this one is being used for serious covid illnesses. But again, the other one wouldn't be used for covid either.