Why can planes take off in the fog (we watched quite a few leaving) but are unable to land? A: The visibility requirement for takeoff is less than for landing.
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With this in mind, horizontal winds (also known as “crosswinds”) in excess of 30-35 kts (about 34-40 mph) are generally prohibitive of take-off and landing. As far as how this happens, it depends on where you are in flight.
For example, the Boeing 737 has a maximum crosswind landing factor of 30 knots (just over 55 mph). So that answers the second part of your question. Yes, a 'plane can land in 50 mph winds.
While rain has a very slim chance of canceling a flight, there can be a minor chance that rain will delay it. Rain is just water, no matter the pressure. Modern aircraft can generate lift regardless of the heaviness of the rain. Planes can and will take off and land in the rain.