my understanding, welding is where the metals mix and bond directly. Solder is where they flow and fill the gaps completely, but they're not mixing. According to the webs welding is where both metals melt and mix. Soldering is where only the solder melts. I think you can imagine that if you flow a solder completely between two piece of metal then there'd be no air and it'd basically a really good cold weld. Imagine sticking two pieces of wood together with peanut butter. The PB doesn't actually chemically bond to the wood nor does it merge with the wood.
Cold welding in my understanding is where the metals are very very flat against each other and held togehter by that. like when you have 2 sheets of paper against each other and try to pull them apart without sliding them off of each other or grabbing corners.
In a good weld you actually make 2 (or 3) pieces of metal into one.
No, you don't need to melt the metal to mix; it's just almost always easier to do it that way.
As soon as you cause conditions under which grain boundaries wander, you get welding if the two pieces happen to touch at an atomic/molecular level.
Note explosion welding: you use a shockwave to hold the pieces together while aggressively dislocating grain boundaries. The result is a (very good) weld. Many glues involve welding behavior, especially if they are used without waiting minutes to hours for the bond to harden before loading it.
For example, "contact cement" (polychloroprene glue) works by precipitating a polychloroprene layer from a solvent into the surface pores of both to-be-bonded parts, letting all the solvent dry, and then forcing such prepared surfaces together to cause intimate interaction of the polymer chains on the surfaces to weld into a single layer of polychloroprene that's solvent-soaked into both it's sides (which wouldn't be possible unless the materials are extremely porous).
However, solids are very bad at wetting surfaces, so you will have a hard time getting the needed atomic contact.
Welding isn't really applicable to composites like wood or paper.
> Friction stir welding (FSW) is a solid-state joining process that uses a non-consumable tool to join two facing workpieces without melting the workpiece material.[1][2] Heat is generated by friction between the rotating tool and the workpiece material, which leads to a softened region near the FSW tool. While the tool is traversed along the joint line, it mechanically intermixes the two pieces of metal, and forges the hot and softened metal by the mechanical pressure, which is applied by the tool, much like joining clay, or dough.
In a good weld you actually make 2 (or 3) pieces of metal into one.