Your traffic seems to have increased around the time of the big Panda update around the 21st August.  Looks like (in contrast with your competitors) you were doing something right.

You might like to keep an eye on this chart: http://www.rankranger.com/rank-risk-index which I find gives a nice overview of the serp fluctuations.

Yes, this can cause a problem as the links will be classed as unnatural.  The best way to get round this is to make sure those links have the rel="nofollow" tag.

The penalty you should be worried about is Penguin, as this will hit pages with overoptimised anchor text links, and a menu link to an external site could generate hundreds, if not thousands of backlinks to another site, so the site you are linking to in that menu could have problems.

There are many instances being posted here of people being hit by penguin because they got featured sidebar links on other sites, resulting in 1000's of backlinks appearing in opensiteexplorer, all with the same optimized anchor text.  I liken the menu links to be similar in result.

But like I said, just ensure that the mass menu links to the other site are nofollowed and you should be ok.

Checking Ahrefs it is clear they have been hit by a penalty.  They have over 700 exact match anchors for "man with a van" for example from just 30 domains, which makes up 67% of their backlinks!