Searching for accurate specification - panel data

#1
Hello Everyone,

I am about to find best specification for my dataset.
To begin with - I am trying to probe the effectivenes of the special economic zones in Poland in the meaning of growth of the economy in three similar panel data models for explained variables: a) registered unemployment rate b) GDP per capita c) gross fixed capital formation per capita.
Data is for NUTS3 sub-regions, explanatory variables are: 0-1 variable for presence of the SEZ in sub-region in year t and a few of the economic variables; yearly frequency, dataset is 2004-2012 for 66 sub-regions.

I have tried fixed and random effects. As for now, I have chosen FE, because of significance and theoretically correct signs. But there are some issues that hold me from taking it as granted:
1. autocorrelation and cross-corelation - how to test it?
2. testing for error term distribution - no idea how to do it in Stata, and furthermore - if it is not normally distributed, should I care about it much?
3. corr. coefficient between explanatory var's and error term - as I have understood from the literature, values near -1 or 1 are not bad as a matter of fact; in my case, its nearly -1 as you can see.
4. what about mixed model? Is it appropriate for my dataset?

I attach outcome for the model explaining unemployment rate.

Code:
xtreg  st_bezr sse01 wartosc_sr_trw_per_capita zatr_przem_bud podm_gosp_na_10tys_ludn proc_ludn_wiek_prod ludnosc_na_km2, fe

Fixed-effects (within) regression               Number of obs      =       594
Group variable: id                              Number of groups   =        66

R-sq:  within  = 0.4427                         Obs per group: min =         9
       between = 0.3479                                        avg =       9.0
       overall = 0.2365                                        max =         9

                                                F(6,522)           =     69.10
corr(u_i, Xb)  = -0.9961                        Prob > F           =    0.0000

-------------------------------------------------------------------------------------------
                  st_bezr |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
--------------------------+----------------------------------------------------------------
                    sse01 |  -1.406066   .4631984    -3.04   0.003    -2.316028   -.4961045
wartosc_sr_trw_per_capita |  -.0000963   .0000166    -5.79   0.000    -.0001289   -.0000636
           zatr_przem_bud |  -26.11989   4.992198    -5.23   0.000    -35.92716   -16.31263
  podm_gosp_na_10tys_ludn |  -.0201788   .0030788    -6.55   0.000    -.0262273   -.0141304
      proc_ludn_wiek_prod |  -229.1996   16.92631   -13.54   0.000    -262.4516   -195.9475
           ludnosc_na_km2 |   .0790167   .0120865     6.54   0.000     .0552726    .1027609
                    _cons |   161.9786   10.76989    15.04   0.000      140.821    183.1363
--------------------------+----------------------------------------------------------------
                  sigma_u |  53.986519
                  sigma_e |  2.5446248
                      rho |  .99778327   (fraction of variance due to u_i)
-------------------------------------------------------------------------------------------
F test that all u_i=0:     F(65, 522) =    27.09             Prob > F = 0.0000
Thank you in advance for any advice :)