It sounds like you have a very sensible and thorough approach to the project. Just a few suggestions:
Saw Blade
Most blades that come with table and miter saws are fairly low end. If you are cutting plywood for finish carpentry, you want a high tooth count blade for smooth cuts. Check out one of the blade manufacturers for the range and you will find some recommended for this type of work. The cost is worth it and it will last through many more projects.
Angle cuts
This is where your cut-twice idea is a good idea. You are just trying to replicate a given angle. Measure the angles either with a protractor, or, if the piece being matched is an angle cut across a short dimension (rather than a long bevel) lay it up against the angled saw blade to check.
Use scrap wood to make a try cut angle based on your best estimate. If it is off, adjust the blade or angle guide slightly and try again. When you have the exact angle, cut the good wood.
If you need to make an angle cut on a wide board on the table saw, consider an adjustable taper guide that runs along the fence.
Materials
It wasn't clear to me whether you were going to make the face frames out of pine or just make mock ups and then use oak. For the actual frames, use oak. It is both more rigid and much less prone to seasonal change. All woods shrink and swell somewhat, even when well sealed, but softwoods much more so. You could find shifting of doors and binding drawers if you use a wood that changes much.
Cutting on/next to marking lines
It doesn't matter. What does matter is consistency. You need to know where your mark is in relation to the exact measurement, and cut accordingly. And do the same thing every time.
For example, if you measure a board using a rigid rule, and you mark with a pencil, you can have the far edge of the pencil line at the exact distance. The thickness of the line is within the length you want. Then make a line across the board using a square at the exact spot. Many woodworkers use a scribing tool instead of a pencil to get a finer marking. In this case, you would cut leaving the line behind, since it was within the length you wanted.
Also realize that when your finished cabinet meets other surfaces, such as the wall, it will almost never be an exact fit, due to slight irregularities in angle and levelness of the wall. This is where trim strips come in. While there is a technique for scribing an exact contour to fit a cabinet to a wall, that is a bit more challenging. Unless you are going to do that type of trimming, leaving a cabinet a hair shallow or short is much better than too deep or too high. Shims and trim work wonders.
Honestly, a 12 year old black and decker probably needs to go to the scrap yard. Low end saws are not built to be rebuildable, at least since the late 20th century (some 50 year old "low end" saws were a lot less "low end" in construction) - a 21st century low-end saw was usually built with "planned obsolescence" from the design phase forward.
Provide the exact model of your saw to check some details.
The one thing (if that's been left serviceable on this model) it might be worth trying is to replace the brushes if they are excessively worn. Beyond that, replacing the saw is far more likely to be cost-effective.
Best Answer
What you just described (cutting multiple pieces at the same time) is in the construction field referred to as "gang cutting". And, yes, if your only power tool is a circular saw, than to get the most precise and exact lengths, cut them in a gang cut. What you should first do is make an initial gang cut at the bottom so as to square the ends. Than measure and mark the length. After they are clamped strike a line with a square across all pieces. If you have a steady hand slowly begin the cut being sure to keep the saw blade to the waste side of the line. Or clamp a guide fence down so that when the saws base plate edge rides along it the blade is at the appropriate mark (usually 3- 3 1/2 inches from inside blade edge to outside plate edge).