I expect the cold water comes straight off the mains. Common practice in some towns.
What colour is your pump? Not yellow, is it? Does it appear to hum softly when running? Picture, please
From what you describe I see no reason to pump the hot water, except for a shower. It would be simpler and quieter to let everything else run on gravity. If flow is poor there must be a defect such as bad pipe layout or constricting valves. If you end up needing a pump, consider getting a pump for the shower only, or a shower incorporating a pump, and bridging the pipework at your old shower position.
Good idea to check the pump filters (if any) although any dirt would usually collect at the bottom of the cylinder and not reach the pump. Apart from fragments of drowned wildlife, you might find limescale flakes if you are in a hard water area (they will also be visible at the bottom of the loft tank).
See if you can get a garden hosepipe conector that will fit on the pipe supplying water to the pump, and see what the flow is like unpumped. It is very likely 3/4" BSP (the same size as a washing machine hose, and you can buy them in various lengths)
Also look at any flexible hose connectors to see if they are twisted or kinked, and at all the service valves to ensure they are not closed, including in the loft and around the pump. Post some pics of the installation please.