Do birds ever fly in clouds?
Yes, they do. 

Pilots normally say that "birds are day VFR only", you're gonna read this a lot on the internet. VFR means Visual Flight Rules, where the pilot operates the aircraft in weather conditions clear enough to allow the pilot to see where the aircraft is going. Thus, they are implying that birds avoid flying at night and inside clouds.

However, this is wrong. Birds do fly inside clouds (and at night), especially migratory birds that have to fly non-stop for weeks or months. They cannot avoid flying at night (of course) and, in several situations, they cannot avoid flying towards a cloud.

According this paper in Science, from Weimerskirch et al. (2016), frigatebirds can even do this on purpose!


  We studied the three-dimensional movements and energetics of great frigate birds (Fregata minor) and showed that they can stay aloft for months during transoceanic flights. To do this, birds track the edge of the doldrums to take advantage of favorable winds and strong convection. Locally, they use a roller-coaster flight, relying on thermals and wind to soar within a 50- to 600-meter altitude band under cumulus clouds and then glide over kilometers at low energy costs. To deal with the local scarcity of clouds and gain longer gliding distances, birds regularly soar inside cumulus clouds to use their strong updraft, and they can reach altitudes of 4000 meters, where freezing conditions occur. (emphasis mine)


Source: Weimerskirch, H., Bishop, C., Jeanniard-du-Dot, T., Prudor, A. and Sachs, G. (2016). Frigate birds track atmospheric conditions over months-long transoceanic flights. Science, 353(6294), pp.74-78.

PS: You have two questions here, specially after your edit. I suggest that you post another question regarding the spatial orientation, since asking different questions (even if they are related) in the same post is not nice, and it's a reason to close ("Too broad: Avoid asking multiple distinct questions at once"). However, it's worth mentioning that apparently birds do suffer from spatial disorientation when there is no visual cue. According to this relatively old paper, "Spatial Disorientation in Birds":


  The only conclusion is that birds are susceptible and suffer from spatial disorientation, and further that the causes of spatial disorientation in birds are exactly the same as those which affect the human pilot, namely; (a) the loss of true visual cues to the horizontal; (b) inexperience in flying under such conditions where visual cues are lost;