Why does my microwave keep throwing the circuit breaker?
This sounds like pure luck that you haven't had problems before. Niall has the ratings right: My microwave says 1100W power output, and 1500W max power use (which is 13A), and a 1/3HP sump pump is 9.5A (or 10A for a 1/2 HP pump), with a peak around 15A when it first starts up. If this is a 20A circuit, you're likely over the limit, and if it's a 15, you're definitely over.

After rain is very easy to explain: your sump pump is coming on more often. The starting surge of sump pump is around 5A, and the microwave probably has a surge of at least 2A when it starts up. 

Basement lights are easy to figure out, just add up the wattage of the bulbs (and current in Amps = # Watts / 115V). Likewise for anything plugged into the basement outlets.

Since the sump pump turns on and off at random times depending on the water level, it's likely that any time it tries to start while you have the microwave going it's tripping the breaker. When the microwave is on a lower power setting, it may not cause the same problem.

If possible, your best bet is to run a new circuit for the microwave: by the sounds of it, it's been retrofitted, and someone just picked the nearest circuit. I would try to find where it was added (assuming unfinished basement or at least access to the ceiling or other relevant areas), and add a junction box and a new line back to the panel, and give it its own 20A circuit. 

Next best is to run a new circuit for the sump pump. You'll still have the issue with anything plugged into the basement outlets, but at least that's more controllable.