Very good advice above. It is always best to see a doctor, in person not on the web, when you have unusual, undiagnosed health issues/symptoms.
In the mean time, stop by you local public blood pressure tester (all pharmacies including those in super markets and Walmart) - check you BP several time, each time try to sit still for a couple of minutes before taking the measure. If you consistently get low reading, you have some hard evidence of the condition, but not an answer on how to address, you need to get advice from a doctor who can check you physically in the office and have blood tests prescribed/done/
If you are passing out then I would just go and get a check up. It is possible you are dehydrated or you tend towards low bp and maybe just need to add a bit of sodium to your diet but it best to go and get a complete blood panel to check for any issues like Thyroidism or electrolyte imbalances as well as maybe getting a stress test or stress echo to see how your heart is responding to exercise. Your issue could very well be something simple but you should probably just go and let a doctor take a look at you. Take care.
It sounds like low blood pressure, which is very common after exercise. The next time it happens, try to move around a little (or if you are about to faint, sit down). When you move around, you prevent blood from being pooled up in your dilated veins (which happens after exercise), you squeeze the blood back to your heart (where it can have useful purposes).
I assume you aren't dizzy/chest pain/fainting DURING exercise. If you are, that's far more serious and should be evaluated ASAP.
The changes in heart rate with breathing is completely normal. The fact that it's more prominent after exercise is also very common and normal. See your doctor, and if you collapse, you should get evaluated fairly quickly.