can you cover a small piece of steel metal with solder so it looks like it has been chromed? Will it stick?

Answer 1

Solder means a wide range of alloys. Steel also means many iron-based alloys.

The classical eutectic electronics (lowest melting temperature) solder (an alloy of about 62% of tin and 38% of lead) will wet and stick to a “typical” (iron with less than 5% carbon) steel. You only need to clean the oxide from the steel surface first, apply sufficient heat, and use an acid-based flux (solder flux also comes in widely different chemical mix) which will eat away the oxides to expose the iron molecules for the tin molecules to wet then the lead will bond to the tin.

When it all cools down you can buff and have a shiny surface. The buffing is to expose the metal grains that have their cloud of electrons giving continuous, not quantized as of an oxidized surface, free valence energy levels which reflect all visible light’s wavelengths of a continuous range of energy levels, and hence be white and highly reflective.

The bad news is that the shining of a buffed solder surface will last only a short while because the surface will quickly get oxidized.

To make a permanently shining surface you have to either (1) keep the surface from ever exposed to oxidizing chemicals (the strongest being fluorine, chlorine, and oxygen), or (2) make the surface not a pure metal but an alloy of sufficient electron negativity. The always-shiny metal sealed in the back side of a mirror exemplifies the former application, and a shiny chromed plated car bumper the latter.

Answer 2

Yes, but it will not look as nice an smooth.
Use solder paste, like is used to attach SMD components to a Printed Circuit Board.
Use an oven to get the steel hot enough to stick. An industial oven, not one used for food.
And use lead-free solder with silver in it. The lead is toxic if someone will touch it or breath it.
If you keep it flat, it might shine, but never like a mirror the way chrome does.