How to calculate wind direction from U and V wind components in R

Question 1

There are three problems with this:

You cannot convert m/s to radians. In order to input wind components into atan2, you must normalize them, but you don't do this by multiplying m/s by pi/180 (which you did to get u_rad and v_rad). You should make a column of absolute windspeed (sqrt(u_ms^2 + v_ms^2)) and take atan2(u_ms/wind_abs, v_ms/wind_abs). (also note that atan2 takes y component first - make sure that's what you want)
atan2 will give you an answer in the unit circle coordinates, which increase counterclockwise and have a zero on the x-axis. You want an answer in cardinal coordinates which increase clockwise and have a zero on the y-axis. To convert unit circle to cardinal coordinates, you must subtract the unit circle angle from 90.
You must know whether the wind info refers to the direction the wind is coming from (standard for cardinal coordinates) or the direction the wind is blowing (standard for trig/vector operations)

If you are given u_ms = = -3.711 and v_ms = -1.471 (on the unit circle it is blowing down and slightly to the left, so it is coming from the northeast), then:

wind_abs = sqrt(u_ms^2 + v_ms^2)
wind_dir_trig_to = atan2(u_ms/wind_abs, v_ms/wind_abs) 
wind_dir_trig_to_degrees = wind_dir_trig_to * 180/pi ## -111.6 degrees

Then you must convert this wind vector to the meteorological convention of the direction the wind is coming from:

wind_dir_trig_from_degrees = wind_dir_trig_to_degrees + 180 ## 68.38 degrees

Then you must convert that angle from "trig" coordinates to cardinal coordinates:

wind_dir_cardinal = 90 - wind_dir_trig_from_degrees
[1] 21.62284 #From the northeast.

Question 2

While the accepted answer has the right idea it has a flaw. As mentioned in the comments one does not need to normalize the u and v component in order to use atan2 on them. The flaw comes when u == v == 0 and wind_abs becomes 0. In C# the two divisions will return infinity (in compliance to IEEE 754) and atan2 will return NaN. When not normalizing the components atan2(0,0) happily returns 0. So not only is normalizing not necessary, it also introduces an error.

Please also be aware that the most common function signature of atan2 is atan2(y, x) -- Microsoft Excel being an exception.

Question 3

In Python:

Dir=np.mod(180+np.rad2deg(np.arctan2(U, V)),360)

This leads to a Southerly wind (180 degrees) for a [0,1] vector, a Northerly wind (0 degrees) for a [0,-1] vector, a South Westerly wind (225 degrees) for a [1,1] vector:

U
Out[86]: 
array([[ 0.  ],
       [ 0.  ],
       [ 1.  ],
       [-3.47]])

V
Out[87]: 
array([[ 1.  ],
       [-1.  ],
       [ 1.  ],
       [-1.47]])

np.mod(180+np.rad2deg(np.arctan2(U, V)),360)
Out[89]: 
array([[180.        ],
       [  0.        ],
       [225.        ],
       [ 67.04097233]])