A carpenter is a person who works with wood. That can either be (for example) making furniture from wood, or making the doors, doorframes, skirting boards etc in a house, or putting together the studs, rafters and floorboards of a house. Some of these tasks are fine and detailed some are not, but the furniture-maker and the rafter-installer are both types of carpenter.
Many carpenters, especially the "put up the rafters" type, will also be skilled builders. They will be able to dig foundations, lay bricks, set concrete. A builder is anyone who is physically involved in building.
Construction worker tends to refer to builders who are working on large projects like skyscrapers. This tends to be a relatively low status job, involving assembling pieces that have been made.
The specific name of a builder who repairs roofs is a "roofer".
The (high status) person who designs a building is an architect.
We did hire an architect to design our dream home, because we wanted to make the best use of the space. The architect also recommended some builders. We didn't want a lot of construction workers with no real skills, so her advice was invaluable. There was Jake who did the carpentry, Jock the bricklayer and Jo who mostly did plastering but helped with the foundations and any other little jobs around the site. We didn't hire a roofer, because our roof was a standard tiled type. Jake and Jock did the roof together. But, of course, we did need an electrician and a plumber, who both were very costly. Our architect also suggested a project manager, who made sure that Jake, Jock, Jo and the others were used efficiently. Unfortuntely we didn't have enough money to hire a garden designer, so we asked Jo to level the garden and turf it. Perhaps next year we will work on the garden.