I am finishing off a basement. I ran 20 amp 12-2 wire with ground for the outlets and 15 amp 14-2 wire with ground for lights.

1) Can I use 15 amp outlets for the 20 amp outlet run? Or does that overload the outlet?
2) What is the functional difference between 15 amp and 20 amp outlets?
3) Can I run a leg off a 20 amp outlet to a remote 15 amp light without any electrical danger? Do I use 12-2 or 14-2 for the leg to the light?

Personally I would only use 12/2 with 20 amp outlets.

1 - - If you use a 15Amp rated outlet on a 20 amp rated line, you must use a 15Amp breaker and limit the electrical load to a max of 15 Amps.

If you use a 20 Amp Breaker with a 12/2 20 Amp wire you should not have any problems with the load on the line as long as the max load is limited to 20 Amps.

2 - - 5 Amps more of safe circuit current.

3 - - Again the limitation is the Amperage draw of the total circuit.

Electricity is a "Demand" product. Just because you put a lamp on a 15 Amp Circuit does not mean the lamp itself will draw 15 Amps every time it is on. If that were true, then the amount of circuits that would need to be run for an average size house would be staggering.

Then there is this little problem of the NEC Code. You must meet the requirements of the NEC Code to be compliant. For example, you cannot use 14/2 to carry 20 Amps. In theory, the wire would overheat and melt the insulation or even shear the line.

Also, you cannot use a standard wall receptacle near water. The Nec Code calls for the use of GFCI outlets near water.

There are several electricians on this site and since I am not one of them, I will defer to any of their responses.

(2) the 20 AMp outlets accecpt 20 A 120 V plugs and 15 A 120 V plugs. On a 15 A receptical both slots are verticle. On the 20 A, one side has both a verticle and a horizontal slot. The 20A outlet can carry 20 Amp safely. The 15A outlet can carry 15 Amps safely.

1) Can I use 15 amp outlets for the 20 amp outlet run? Or does that overload the outlet?
Table 210.21(B)(3) in the NEC allows a circuit rating of 20 amps to use 15- or 20-amp receptacles. From the commentary”…two or more 15 amp receptacles or duplex receptacles are permitted in a 20 amp general purpose branch circuit.” This actually derates the circuit based on the 125% rule. The breaker rating doesn't overload the outlet, the total current draw of the item(s) plugged in the outlet overloads it.

2) What is the functional difference between 15 amp and 20 amp outlets?
See KISS explanation.

3) Can I run a leg off a 20 amp outlet to a remote 15 amp light without any electrical danger? Do I use 12-2 or 14-2 for the leg to the light?

15 and 20 amp branch circuits can supply lighting, but you must use 12-2 to the lights. NEC 210.23 and 210.24 set the requirements. If the circuit supply's both a light and two or more fixture outlets, then the circuit must have the same rating as the breaker-if using a 20 amp breaker all branch circuit wire must be #12.