You're absolutely right, cats can also develop diabetes after a Depo injection. However, increased thirst and urination are also side effects of steroid administration. In other words, the increased urination may go away on its own once the drug wears off. Your vet can easily check his blood glucose to look for evidence of diabetes this morning.
I cannot say for sure just from your description that your cat has a heart problem, but I am concerned. Cats with this issue never do show signs prior to the Depo shot - otherwise we wouldn't give them the injection. The problem occurs (we think) because Depo causes the volume of the blood to increase. This increases the work load on the defective heart. The heart was previously able to compensate for the problem, but with the additional blood and additional strain, the cat begins to show clinical signs of a problem.
Again, you're right. The increase in stress from travel and the visit to the vet could worsen a heart condition. Unfortunately, I think it is a necessary risk in order to diagnose and treat your cat. Signs of a problem would be discharge from the nose or mouth (usually clear or pink-tinged), coughing, difficulty breathing, pale or bluish-gray gums and lethargy. The only thing you can do at that point is get him to the vet as quickly as possible so that he can be put on oxygen and given injectable medications to help remove fluid from his lungs. Remember - we are still just hypothesizing that he has a heart condition.
I agree, the drug has its serious downsides. We do our best to educate clients before using it that there are possible side effects. We always have to weigh the risks with the potential benefits for each cat. With some cats, there is no other option to manage conditions requiring steroids - as so many cats are difficult or impossible for the owner to orally medicate.