Dealing with very small samples with different shapes I have two groups with very small sample sizes (just 6 obs per each group).It seems that There is a violation of normality and Homogeneity of variances (based on Plots). 
Now I was wondering if I can use  Mann-Whitney test to check the equality of medians although these two group donot have the same shape? Also as levene test doenot have the high power when we have small sample size, I though maybe I should compare the equality of variances based on rule of Thumb,  But I am not sure whether the critical value for the rule of Thumb (equality of variances) is 3 or 9?
Any Advice would be highly appreciated.
This is the first analyte (More information: Both P values based on T-test and Mann-Whitney tests are not significant, T-test P value is 0.2 and Mann-Whitney test is 0.5)

This is the second analyte (More information: Both P-values based on T-test and Mann-Whitney tests are very small, around 0.002)

 A: A permutation test may be best for the first analyte. I don't have your exact
data, so I will use the data below (roughly approximated from your graphs).
I'm using the difference in group means as the metric. The P-value is 0.3.
Among $m = 100,000$ permutations of group labels there were 94 distinct differences in means.
[This P-value is essentially the same as the P-value of a 2-sided Welch t test.]
set.seed(829)
x1 = c(111,113,116,116,119,129);  x2 = c(105,112,125,125,138,142);  all=c(x1,x2)
gp = rep(1:2, each=6)
a1 = mean(x1);  a2 = mean(x2);  d.obs = a1 - a2
m = 10^5;  d.prm = numeric(m)
for(i in 1:m) {
  prm = sample(gp)  # randomly permutes groups
  d.prm[i] = mean(all[prm==1]) - mean(all[prm==2])  }
mean(abs(d.prm) >= abs(d.obs))
[1] 0.29665
length(unique(d.prm))
[1] 94

hdr = "Simulated Permutation Dist'n of Diff in Gp Means"
hist(d.prm, prob=T, col="skyblue2", main=hdr)
abline(v=c(-d.obs, d.obs), col="red", lwd=2, lty="dashed")


t.test(all~gp)

       Welch Two Sample t-test

data:  all by gp
t = -1.1207, df = 6.8904, p-value = 0.3
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -22.337040   8.003706
sample estimates:
mean in group 1 mean in group 2 
       117.3333        124.5000 

