1. You need to work with your vet on priortizing the tests due to the costs. If the lungs sound really severe, the question is, does your vet want to do a chest x-ray first, or maybe consider a basic blood profile for an overall health assessment.
2. An antibiotic shot is worthless, unless it is convenia which lasts for 14 days.
Your pet would really need to be on a good oral antibiotics, especially, with a cat fight.
3. The unsteady on the feet is very concerning, and suggests to me there may be other underlying health issues that need to be addressed. In that case, at least a basic blood test is critical to help understand what additional treatment might be helpful.
4. There are medications that can help with heart disease and congestive heart failure, but only your vet can make the distinction on examination of these drugs should be started now or not.
I wish I can offer more over-the-counter options, but that's the best advice I can give under the circumstances.