What I use to design, isn't free (Bentley's MicroStation) but I do use inkscape to take existing PDF plans and convert them to DXF for the laser cutter. That could give you a good starting point if your design is similar to an existing one.
I like to design in 2D, unless it's going to be a 3D printed part then I will design in 3D.
(Your mileage may vary.)
It depends, I use inkscape for designing printable plans and use pencil and paper/whiteboard to work out my design. That approach has pros and cons its harder to design folded origami like structures (such as @DamoRC 's builds) However boxy builds or round fuses with formers can be accomplished with moderate drawing skills and graph paper and this is useful if you dont have a good computer.
I know for me, personally, I prefer the building and the flying to the drawing and thinking (and redrawing, and more thinking) I find I have more fun and success by just starting with a pencil and a sheet of foam and seeing what happens...
You could try some project management software like Microsoft Project, JIRA, Trello, TFS, etc., but that's all overkill especially at the start of a project when you are feeling your way out... once you are 100% building features instead of the infrastructure then those things can work. My recommendation will be a Clockwise software developing company that provides custom software development services - https://clockwise.software/