I am a licensed electrician, and your panel, though in pretty ratty old shape, is probably still safe. I would absolutely upgrade if it were mine, but that's your call. I will also not comment on your circuit design. I would not build that design either (underpowered, overworked) But again, that's your call. If you have appropriate fuse protection, the worst that would happen is popped fuses ( a lot of them probably).
However, the drawing you have made shows one wire going into the receptacle and one wire coming out (same on each side)- I assume you mean incoming wire under one screw and outgoing wire under the other (same on both sides) This is not the best way to wire a receptacle. Each color wire should always have a short piece of wire attached with a wirenut (a bundle of three wires under a wirenut). This short piece then attaches to the receptacle. That means one black wire under colored screw and one white wire under the silver screw. This method is best for two reasons. First, the little tab between the two colored or silver screws is not meant to carry a full circuit load. See how small the little tab really is? It will get hot if the other receptacles are loaded. It is there so a duplex receptacle can be split into two separate outlets. Example: top outlet switched, bottom outlet hot all the time (with appropriate extra wiring). The other reason is that if a particular receptacle fails, the entire circuit is not compromised, and the branch wiring may not be burned up. Only the short tap wire will fry.
If you do it the way it's drawn, you will certainly not be alone. Many people make the mistake, and many people get away with it forever. However, it takes little extra time to tap the branch wires, and it will be correct, and safer, by any account.
Easy to do, harder to write.