HELP: un paired t test what do these results mean?

 A two sample t-test with equal variances and sample sizes (which is what you have) has a t value of 13.63.  The only problem that I can see is that the DF for the test should be 12 and not 11 (n1 +n2 -2) = 7+7-2=12. The results of your t-test, for the samples taken, is that there is a significant difference between the two means.  An examination of the two standard deviations – using the F test, indicates there is not a significant difference between the variances. 
  If your data is representative, this would suggest that there really is a significant difference in the inventory levels and that the “noise” in both inventory systems is about the same.  Assuming that these average differences matter and that you wish to make inventory levels the same, this would suggest that you look at things in your inventory process that impact the quantity of inventory and not the changes (variability) in the inventory quantity.

No, it shouldn’t make any difference.  However, the missing August data does explain why you have DF = 11 and not 12 since you now have (7+6-2) = 11.
  Since you have monthly data I would also recommend that you plot both inventories on the same graph by month and take a look at the pattern.  You may find something of interest with respect to trending over time – i.e. similar trends but different levels, inverse trending – in a month when one goes up the other goes down, etc.  Such trends my give you even more insight into your process and may be far more valuable than the simple fact of the statistically significant difference of their means.

With such small sample sizes (n=7 for each variable), wouldn’t a non-parametric test (like Mann-Whitney) be more appropriate? 

Right or wrong, here are the assumtions I would check before using a t-test:
1.  Data stability (if it is time ordered, which is not the case in this example)
2.  Data normality – I know that t-tests can be robust against non-normality, but I was under the impression that this robustness was linked to sample size (with larger samples sizes being more robust).
3.  Test for equal variances
Given that we only have 7 data points for each time period, I’d be jumpy about the validity of the normality test and the test for equal variances.  Instead of straining to use a tool where I can’t really tell if its assumptions are met, I would take the “easy way out” and use a different tool where I KNOW the assumptions are met. 
With all that being said, based on the output from the t-test I think a nonparametric test would have given the same result (“is there a significant difference – why, yes, there is!”).