What no. Opamps don’t multiply and they don’t operate at the speed of light. They have some timescale that goes like their bandwidth, which depends on their feedback path.
Yes, feedback op-amps definitely have bandwidth limits. Although you can get ones in the gigahertz range now.
Analog multiplier ICs are available.[1] They're not common, and they cost $10-$20 each. Error is about 2% worst case for that one. There are several clever tricks used to multiply. See "Gilbert Cell" and "quarter square multiplier".
This is absolutely not true, the speed of analog circuits is (by a significant margin) determined by parasitic capacitance, inductance, and resistance of the components.
To put numbers to it, a typical high performance analog multiplier might have a loop length of 1cm for the feedback path. This circuit should theoretically operate at 30GHz, but realistically such circuits operate with a bandwidth measured in megahertz.
