Non-coherent receiver for DBPSK: Implementation details
1) A typical approach would be to have a frame alignment word (and by "word" I mean bit sequence) at the beginning of every frame.  You could time synchronize by looking for that FAW through cross-correlations.  Barker codes are a good choice because they have good autocorrelation properties.

The problem with the approach is that the data, being random, can look like the FAW sometimes.  You can use the periodicity of the FAW to screen out those false positives.  For instance, if your frame is 1000 bits long, you would look for multiple hits spaced 1000 bits apart.  Once you have found them you have both time and frame synchronization.

2) There is not a direct relationship between carrier frequency and bandwidth.  There is a direct relationship between bit rate and bandwidth, but there are other factors involved.  The easiest way to determine your bandwidth is to take an FFT of your signal and measure it.

3) Your question is unclear.

4) You haven't provided enough information, and "optimal" can mean a lot of different things.  If you want a good answer to this you should start a new thread with more details.

5) It sounds like you are adding noise via AWGN, so you know what your SNR is.  With BPSK Eb/No is equal to the SNR.