Need some electronics help!

[Tenacious]

The corrosion looks like it is from water with the power ON.  You have probably already tried simply cleaning all of the corrosion with a fast drying, non-residue solvent and a tooth brush.  Try to work the bristles behind the pins and under the chip to get it all.  All of that fuzz may be conducting small currents between the pins of the chip that aren't supposed to be there.

If cleaning is not enough, I would definitely replace the chip first and wait and see about the capacitors.

[Tenacious]

Just one question, can the voltage a mutimeter supplies in continuity mode (resistance mode) damage a component?  I'm assuming not, as I think it is very low (can't remember exactly what batter this one has in it).

Not in a power supply circuit, like yours.  TCMSLP gave a pretty good description of what you should see testing for the various components.  Sometimes you can "lift" only one leg of a 2 legged component, without removing it entirely for test.

You can damage the circuit under test IF it is powered and your meter is in current mode while you are trying to read voltages.

You can damage the meter IF the circuit is powered and the meter is in resistance mode.

In general, if electronics get wet, but, are not powered ON until after completely drying out (and conductive residue removed), they will usually survive.  If powered ON while still wet, destructive currents can have pathways to sensitive components far from the power supply, wiping out many components.