Phase and Amplitude estimation
answer to 1, No,  the  phase of an tone isn't constant from one FFT to the next, so averaging phase provides no reduction of error unless there is reason to think there is like in a coherent algorithm like SAR. 

Answer to 2 is a bit more complicated but is yes, because you need to do carrier and timing recovery in a communications systems

For something less structured There are several frequency estimation algorithms that work on FFT data like:


  Macleod, Malcolm D. "Fast nearly ML estimation of the parameters of
  real or complex single tones or resolved multiple tones." IEEE
  Transactions on Signal processing 46.1 (1998): 141-148.


and this just a personal favorite. There are 270 papers that cite that paper and many before that.  This method is nearly ML, so not Welch but not shabby.


  La Scala, Barbara F., and Robert R. Bitmead. "Design of an extended
  Kalman filter frequency tracker." IEEE Transactions on Signal
  Processing 44.3 (1996): 739-742.


Is more along the lines of tracking frequency and there are 170 papers that cite this paper.

a good book is:


  Quinn, Barry G., and Edward James Hannan. The estimation and tracking
  of frequency. Vol. 9. Cambridge University Press, 2001.


A recent tutorial paper 

Vila-Valls, Jordi, et al. "Are PLLs dead? A tutorial on Kalman filter-based techniques for digital carrier synchronization." IEEE Aerospace and Electronic Systems Magazine 32.7 (2017): 28-45.

might be useful.

Phase introduces complications because phase is relative to some reference and having the same reference for multiple tones may or may not be meaningful.  

One of the regular contributors here, Peter K has a web page (which I don't recall where) that has a lot of material that may be of interest.