Determining the uncertainty of an autocorrelation
As far as I can see the main question is: How to account for the uncertainty in the measurements of $X_t$ (i.e. $F$).

One approach to do this would be to run some sort of Monte Carlo simulation on the data. I.e. for each experiment you can resample a large number of times (say 1000 or 10000) where each point has $1-F$ probability of being flipped. By calculating the auto-correlation of each resample you get a distribution of the correlation for each $k$. Then take the 95% (or another value) confidence interval to get an uncertainty in $R(k)$.

It may also be possible to do this analytically but I'm not sure for binary data (my instinct says not or at least not simply).

This uncertainty in $R$ should be accounted for in your fitting of the data when calculating the decay time. This can definitely be done analytically and may be implemented in whatever you are using to do the fit (I think matlab does). Alternatively a similar Monte Carlo simulation could be done but is probably more effort.

If you think all the data follows the same trend (not unreasonable if it is a repeated experiment) you could calculate a mean for each auto-correlation distance as you suggest. This is effectively taking a manual Monte Carlo with 25 resamples. Then you can calculate the standard error as you suggest (note: $n=25$, not $5000$). I would shy away from this approach as it ignores the uncertainty in you initial measurement which you presumably have a good reason for giving.