I noticed that since I've started increasing my physical activity significantly, my heart rate has decreased as well. Over the past six months, my resting heart rate went from 88 to 66. I thought this was a good thing, but my weight is increasing despite a strict low calorie diet, so I am wondering if my metabolism has taken a nose dive due to my lower heart rate...
The short answer is no, your metabolism isn't really driven by your heart rate.
The basic definition of metabolism is that it is the process by which your body converts the food and liquid that you consume into energy to fuel the body. This can be generally divided into two components, catabolism, which is the breaking down of ingested items into energy, and anabolism, which is the building up of cells with proteins and nucleic acids. The link I provided gives a basic definition and a basic writeup relating metabolism to weight loss.
The heart has a role in this, in that it delivers oxygen to the body which is used in metabolic processes, but part of the adaptation to exercise is that the cardiovascular system gets more efficient at delivering oxygen where and when needed. This is (in part) why your heart rate slows as you get fit. These adaptations include neovascularization (more capillaries/blood vessels grow into working muscles), stronger pumping action, and ventricular enlargement so the heart can pump more blood per beat. Oxygen delivery remains stable and adapts to deliver what is needed according to activity.
Your most likely culprit for the weight gain is that you are exchanging muscle for fat. Muscle is denser and weighs more than fat does, so if you lose a proportion of fat and replace it with an equal proportion of muscle, you will weigh more at the end of that process.