Fixed effects versus random effects in panel data for intervention group only? Change in dependent variable for each time period

I have a panel dataset covering observations from the same individuals over six time periods (unbalance panel). I am interesting in estimating the change in a continuous dependent variable (i.e. body mass index – BMI) for each time period.

My reading of the literature led me to Fixed Effects and Random Effects. Intuitively I thought FE would work best given I know I have omitted variables (e.g. income, education, genetic disposition), which are time-invariant as is their impact on the individual (given the individuals are 65+ I presume income and education will stay the same).

BMI = Intercept + B1(Age) + B2(Municipality) + B3(Measurement Period), where Measurement period is a categorical variable.

When I run this in R using FE, I get statistically significant results that make sense. However, I have queries as to whether this is the right approach given:

1. When I perform a Hausman Test comparing FE vs RE I cannot reject
   the null, which indicates RE is more suitable for my analysis,
   however, intuitively I thought FE is better. Is it better to go with
   intuition or test results? My research into this is inconclusive.
2. When using a FE regression I get a coefficient for each time period,
   except for the first time period (as this is dropped), for example,
   the coefficient for measurement period 4 = -0.507. Am I correct in
   saying that this means compared to measurement 1, BMI in measurement
   4 is 0.507 lower?
3. Finally, if I want to get the change in BMI from one period to the
   next (i.e. change in BMI from measurement 1 to 2, 2 to 3, 3, to 4
   etc.) what would be the best method to do this? For example, could I
   run separate regressions with different measurement periods as the
   reference? (however, this seems very time consuming).

Any feedback would be greatly appreciated!