We renovated our home last summer (2007); renovated the main-bathroom with new fixtures (sink, tub, shower, toilet, etc.); renovated the kitchen with all new cabinetry, and appliances, including a dishwasher and garbage disposal, both of which we did not have in our kitchen before the renovation. Over the past few weeks, we have begun to have problems with our plumbing. Our dishwasher has overflowed twice - the water did not drain out between cycles. We read our owners manual and it was suggested/recommended to check the water line connection to the dishwasher for "kinks" in the line. We did, and there was a "kink"; the water pressure from our new kitchen sink is "minimal" to say the least. This is particularly alarming considering that when the sink/faucet was first installed the water pressure mimicked a roaring waterfall! In our main bathroom, after flushing, the toilet's water tank does not refill on its own. Now we literally have to reach down and turn the water shut-off-value nozzle all the way off, then turn it back on, in order for the water tank to refill. And now we've also noticed that the pipes in our main bathroom rattled loudly momentarily, followed immediately be a significant drop in water pressure (the hot-water side of the faucet) when we turn the water on full force in the bathroom sink. Any ideal as to what may be causing our problems; are these issues/problems connected/related; most importantly, are these issues/problems repairable and if so, what's our first step in doing so?

Sounds like you have debris caught in your mixing valves and aerator. When was the water heater last flushed? I think everything described is related to debris. Remove the aerator from your kitchen sink and tell me what you find in the plates and screen.

Our dishwasher has overflowed twice - the water did not drain out between cycles. We read our owners manual and it was suggested/recommended to check the water line connection to the dishwasher for "kinks" in the line. We did, and there was a "kink"; the water pressure from our new kitchen sink is "minimal" to say the least. The manufacture meant the discharge hose, not the pressure line.

In our main bathroom, after flushing, the toilet's water tank does not refill on its own. Now we literally have to reach down and turn the water shut-off-value nozzle all the way off, then turn it back on, in order for the water tank to refill. Let us know what type of ballcocks that you have,(see images).

Any ideal as to what may be causing our problems; are these issues/problems connected/related; most importantly, are these issues/problems repairable and if so, what's our first step in doing so?
Yes, this is reparable by the homeowner. Bob was bang on when he mentioned debris. What has happened here is that in replacing all that stuff you disturber the mine4rals built up in your pipe walls and now they're clogging your system. Let's start with the water heater. Do you flush out the minerals built up in it on a regular schedule? If not let me show you how. For long life and fewer troubles you should keep your heater clear of mineral build-up by FLUSHING NOT DRAINING on a regular schedule. Attach a hose to the boiler drain at the bottom of the tank. With the pressure on, open the boiler drain and let it run until the water runs clear. You will see a spurt of red,(rust) followed by white or yellow grains,(lime or calcium carbonate). This shouldn't take more then a few minutes. Do this monthly to keep it clear. Now flush out your water lines on ALL fixtures that are affected and pull each aerator and clean the screens. Be sure you put them back togather the same way you took them out. If a faucet still has a problem after cleaning the aerator you will have to open up the faucet and check and clean the inlet ports of the cartridge. Then nturn the water back on for a few seconds to flush out the supplies and open valve body. Do this on all faucets affected. Don't forget to flush it out every month. Your heater will thank you for it. Good luck, tom