What is the chemistry behind amalgamation?
An amalgam is simply an alloy of some metal with mercury.

Alloys are simply solid solutions with metals. Since the mixture of multiple elements disrupts the metal bonding, the properties are often quite different than the components. (In some cases, alloys form so-called intermetallic compounds with defined stoichiometry, but more often, the ratios are very flexible.)

I wouldn't describe the mixture as a reaction, in much the same way that I wouldn't really call dissolving sodium chloride in water as a reaction. You're forming a mixture. Water disrupts the ionic bonding in NaCl, and mercury and other metals can also accept other components or impurities.

Gallium does alloy with a lot of metals, and indium-gallium eutectic alloy is a liquid at room temperature, as is "galinstan".

As to your question about mercury dissolving the metal atoms, consider that mercury is also a metal, so it also participates in similar metal bonding.