Yes, an aircraft can stay in the air without going forward if the oncoming wind, called headwinds are equal or greater than th minimum speed of that aircraft.
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Because headwind increases the lift, pilots prefer to land and take off in headwind. Tailwind is wind blowing from behind the aircraft. It reduces the lift and aircraft generally avoid taking off or landing in tailwind.
A crosswind above about 40mph and tailwind above 10mph can start to cause problems and stop commercial jets taking off and landing. It can sometimes be too windy to take-off or land. The limitations are in place for the safety of the passengers and crew.
Yes, a 'plane can land in 50 mph winds. Specifically, different airlines will set the parameters for maximum crosswind factors flight crews can land in and not all airlines have the same criteria. The B-52 can. The wings are so long it could easily hit the ground in a crosswind landing.
By landing into a headwind, the airframe is pushed back and slows down faster, allowing pilots to reach taxiing speeds much faster and without wearing out their brakes early. Moreover, the landing speed is also reduced, allowing for a safer entry off the runway.
Headwinds are what work against an aircraft because these winds blow against an aircraft's flight path. Planes flying against headwinds are essentially slowed down by that wind. Think about running into the wind: it's much harder to run into the wind than with the wind (more on this later).
The airplane will still fly; it just takes considerably more runway to depart. Most Pilot's Operating Handbooks (POHs) will use a phraseology in their takeoff roll performance charts such as increase ground roll 10% for every 2 kts of tailwind or similar.
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. The only real problem with heavy rainfall is the decrease in visibility for the pilots.
Turbulence is a sudden and sometimes violent shift in airflow. Those irregular motions in the atmosphere create air currents that can cause passengers on an airplane to experience annoying bumps during a flight, or it can be severe enough to throw an airplane out of control. (The pilots) aren't scared at all.
Severe turbulence can cause a plane to drop so suddenly that pilots temporarily lose control. But, again, that's not enough to crash the plane. That's not to say it's never happened. In 1966, human error and turbulence combined to bring a plane down over Mount Fuji.
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.
Jet aircraft can safely fly over thunderstorms only if their flight altitude is well above the turbulent cloud tops. The most intense and turbulent storms are often the tallest storms, so en route flights always seek to go around them.
With reasonable proficiency, most private pilots can handle surface winds of up to about 20 miles per hour. However, the direction makes a lot of difference, and flight instructors find that one of the most difficult lessons to teach is crosswind landings.
Expect to see many pilots divert in winds much above 40 knots even if the airport is open, especially if gusting to above 50. Crosswinds are a little different as they depend on the braking action - basically how grippy the runway is.