To answer the question asked, whether you can solder 18ga to 12ga and heat shrink the joint, the answer is yes. The industry does it all the time, and there are many places in a standard auto harness where such joints are made. They are usually covered with tape or a loom, so you don't see them, but they are there. The wire used needs to be sufficient for the intended load. If 18ga is large enough to handle maximum anticipated load, you can do what you suggested by splicing to the 12ga pigtails on the plug.
Your other question is whether you can split the wire and use only one fuse. Again the answer is yes, as long as the fuse is light enough to work as a fuse for whatever is on one or the other of the wires, and not rated to handle the load on both wires (unless the fuse and wires are well within the working range of the total load). In other words, if you have two systems running off the same source, and both are, say, 10 amp max loads, you could use a 10 amp fuse before the split. However, in doing this, you would not be able to use both systems at full rating because you would overload the fuse. If you use a 20 amp fuse to protect both, you run the risk of the fuse being too large for the wire, which could be a heat problem or a fire hazard. Better to use two fuses, one in each line, rated to the anticipated load, but you will still need to verify that the max load on both systems is within the handling range of the 18ga wire you want to use, if the wire used up to the point of the split is also 18ga.