Calculating forward voltage for a string of fairy lights: multimeter shows "1"
The way these fairy lights work is how you described, all in parallel. They typically have a single resistor at the battery box if 3 batteries, or maybe no resistor if 2 batteries. Smarter ones with a timer circuit will still be fairly similar. While transistors answer is correct for finding the forward voltage of a single led, at an assumed 20mA forward current, it doesn't answer most of your questions.

To find the voltage and current of the string, you take part of what transistor said, but apply it differently. Use a set of new, good batteries. You will need to measure their voltage while the leds are on. And you would need to take the case apart to access the resistor inside. Measure the voltage across the resistor. This will give you a fairly accurate forward voltage of the parallel led string (Source Voltage - Resistor Voltage = load voltage)

Now remove the batteries and measure the value of the resistor, or read the color code or smd resistor code if available. With this, you have both the voltage across The resistor and the resistance, so you can find the current through it. Voltage / Resistance = Current in amps. Alternatively you can simply measure the current through the circuit with the batteries. Any timer IC is unlikely to consume enough to make the current you measure inaccurate.

Now that you have the total current, you can divide by the number of leds, for a fairly close forward current. You can then use those numbers to replace the resistor and battery box with a usb source and resistor for the usb 5V voltage.

As to your question on supplying or drawing current, USB by standard is supposed to limit to 100 ma without enumeration,  but it is rarely enforced by hardware. You can assume you can pull 500mA or more in 99% of supplies. Usb supplies like wall chargers are often 2.1 amp.  Your load, especially a simple led + resistor circuit, will only pull what it needs.

Your assumption of 3V is a pretty good assumption in practice, as that's less than the Vf of your typical blue or white leds at the recommended nominal 20mA If. They would draw less current and last longer that way. Red leds you should assume lower.

All that said, the easiest thing to get these on a usb supply is to use 1 or 2 diodes like the 1n4001 with a 0.7V average forward voltage drop to drop the 5V usb supply voltage to about the same as the batteries that the fairy light uses. Connect to the battery leads with the right polarity and you are done.