Recommended data cleaning techniques for multivariate time series prediction?
3 questions:


Does it make sense to transform the data into a stationary time series when using LSTM? 
Always Stationarity is always desired property and data should be transformed (read more)
Should the data be normally distributed when you are using PCA? No 
There are multiple assumptions around PCA IF you use some matrix factorization techniques. For example if using SVD you should make sure that your matrix is of full rang.
Does it make sense to impute/fill (by forward filling or something else) big gaps or is it just better in that case to ignore the missing data completely? If for certain features a lot of data is missing you should drop them all together. Dont try to impute it, you will add false information.