Suppose 5 Amps of current runs from point A to C.(20V) I want to connect a thin wire(I believe not for more than 1Amp) to a point B in between A and C. Will the amps melt the wire where it is connected to 5 amps wire? The wire will only be powering 350 milliamps load.
If the load only draws 350mA then that is what the wire will carry.
The potential problem arises that if the 5A circuit is fused at current that is much higher than the thin wire can handle without overheating, you could potentially have a fire or a burn if the thin wire becomes shorted. So generally it's "better" to keep the wires similar size or separately fuse any taps off of a bus bar or thick wire right where it is tapped off.