I am no expert and I am a fairly new diver myself. (3 yrs diving with only 129 dives) But from what I have learned and heard here is my $0.02 worth.
Far_X:
So questions to y'all:
a) What is the ideal breathing rate with minimal activity underwater? How many seconds for one cycle of breathing (inhale/exhale)?
I have no idea. I try not to think about breathing. When I dont think about it, I breath naturally and will usually have an SAC (Surface Air Consumption Rate, which is a formula to adjust your air used to as if you had been at the surface breathing. It allows us to talk about air consumption with out having to factor in depth) of 0.43-0.45 (cubic feet per minute). When I find myself thinking about my breathing it is usually because I am not as mellow as I normally am (mask leaking, or something else causing me discomfort) and I am taking bigger breaths and forcefully blowing them out. SAC then goes to 0.5-0.57. My best ever SAC was 0.37 but that was an exceptionally mellow dive and I quite literally spent a lot of time just sitting and enjoying the ocean.
Far_X:
b) Does increasing the psi decrease the volume of air used per breath or is it just the same?..
The pressure of the air you receive from your second stage (into your mouth) is dependent upon the depth at the time the breath is taken. The first stage regulates the pressure coming out of the tank. At surface it reduces the pressure from 2500 PSI to what is the equivalent of 1ATA (1 Atmosphere of air pressure or normal air pressure at sea level) and at 33 FSW it delivers the air at 2 ATA (or 2x the normal sea level pressure) and so on the deeper you go.
Air consumption is a function of several factors. A search here will reveal several discussions on this as it is a common question. In general here are the items that effect your air consumption:
1. Depth. You will get longer out of a tank diving at 33 Feet or less than you will diving at 66 feet or 100 feet etc. Deeper means less time on the bottom due to increased air consumption (pressure) and of course the Non Decompression Limits too.
2. Proper Weighting. If you are too heavy, you will have to add a lot of air to your BC to compensate. This is of course air you cannot breath and then you will also have more drag with the more fully inflated BC and that means more energy to move in the water and that equals more air used too.
3. Master Neutral Buoyancy. On my dive this weekend I was standing upright about 15 feet from the bottom and 20 feet from the surface watching my dive partner. He was in constant motion, with his feet and hands waving about trying to get a picture. He dives a high pressure (3500 PSI) steel, I dive an AL80 tank. I normally have several hundred pounds of air more than he does at the end of the dive. You will dramatically improve your air consumption when you have mastered your neutral buoyancy. A lot of air is wasted inflating and deflating your BC trying to get neutral. If you are doing the yo-yo routine, bouncing off the bottom and then heading for the surface with a 20 foot swing between, then a lot of air gets wasted. Ad as short of a psst of air as you can, then WAIT. If does take a little time for the added air to affect your buoyancy. Wait about 30 seconds before you add more. Dumping air is instantaneous.
4. Streamline your gear. Dont have stuff hanging down to damage the reef and create drag.
5. Trim. Adjust your weights so that you are swimming horizontal, If you are swimming with your body higher than your feet you are creating more surface area and drag which of course requires more energy to move through the water.
6. Park your hands. Dont use your hands to swim. Your hands should be stowed someplace. I like under my bank at my derriere. Some place them folded at their waist. You should only move your hands to check your gauges or adjust something or maybe use your light. Nothing else. Hands are not efficient swimmers and waste energy which translates to air.
7. SLOW DOWN. Move very slowly. What is the hurry, you got someplace to go? Imitate the fish, they move very slowly unless in a flight mode and then that is a short sprint because speed waste energy. By moving slowly you will do two things A: Increase your bottom time by using less air and B: See more stuff you were just swimming past before with out seeing. If you dont chase the fish, they will get curious and come to you.
8. RELAX. Your breathing should be natural. Do you blow hard to exhale normally? Then dont do it diving. Do you take as big a breath as you can on land? Then dont do in in the water. Big breaths are going to cause you to rise. That is fine if you are using your lungs (NEVER hold you breath, keep that airway open)to make minor adjustments to your buoyancy. But if that is not what you are dong, then you will be adjusting your BC to compensate, and then when you exhale, you will be adding air again. Diving is a Zen thing. I have had dive partners check to see if I was awake because I was so mellow.
9. Be physically fit. From you post you are working on it.
Last of all, is enjoy the dive.