My experience so far has been that Najma souq is an OK place to get hardware, but for do-it-yourself projects, any local hardware store (just keep your eyes open, there are several) will have a similar range of tools, etc; for that matter, Carrefour has a decent tool selection.

Construction-grade wood can be found fairly easily either at Najma souq or in the industrial area (there's one place on Street 5, at least, which has a good selection and slightly better prices than at Najma). Some of the "construction grade" plywood actually has hardwood faces, which can be acceptable for some kinds of furniture.

Laminated pressed wood (like for Ikea furniture) or MDF can be purchased at Najma souq or at one or two shops on Industrial Street 11 (near Al Manajer street), as well as some edging that matches some of the laminate surfaces.

Actual "craft" woods - pine, oak, cedar, mahogany, etc., I have yet to really find. The place on Industrial Street 5 had some large chunks of mahogany, I think, but much too large for my hand tools. I'd love to find milled pieces of wood to use on the edge of the plywood that I've bought, but haven't had any luck yet.