Go ahead nad 301, SE see it as a canonical as index,html is not always the default page. make sure you dont have any internal links point t the .html, of you will leak link juice though the 301 un-necessary.

doing a 301 should consolidate you link juice.

Thwe SEO Toolkit sees the same probnlems as Bing sees, you need windows and you need to install IIS  (add features) first
http://www.iis.net/download/SEOToolkit

sometimes it is better to get a new domain and start again. And do not 301 old domain to your new one.
There are things to consider before doing so, but it sounds like to me your links are crappy,, and who knows what else they have done.

Not quite sure I understand what you want to check, but as long as one 301's to the other it does not really matter. it may take some time for SE's to catch up.

Are you saying you want to check if it is resolving correctly?
in IE click F12, and then select Network and start capturing, you will see if its useing a 301, or a useless 302.

If you want to prove to your client that the developer is not on the ball, do a scan with the SEO Toolkit and show the results, if you dont have windows too install it on, i will do one for you.

Zsolt makes some good points

It not a problem, it’s a natural thing to do.
As far as PageRank goes if you link to every website from every website it will have neutral effect over all, but your worse ranking sites will get a boost at the expense of the higher ranking sites. You can alter your links to benifit a site you want to give a boost

But since there is value in link text relevance, you should come out on top all over
Having said that it’s better to link in a way that looks natural to your users.

I would not have these as my only links if they are on the same ip number, but as long as you have a good range of links from other places it is ok to have links from the same ip number. i think people get a bit excited about same IP or same C-class.

First I mast sure you understand, that no-follow still leaks link juice, it just does not pass it to the linked page.
there was a time where you could stop leaking link juice by using no-follow, but not any more.
So using no-follow’s is generally not a good idea. If you do you are wasting link juice. Even if you don’t want a page to rank, you are better off letting the juice flow and have a link on the linked to page pointing back to your home page or any other page you want to rank.
As for no-follow and the fact that Google still follows. They don’t follow though that link, but they may get to the page from another link, or they may already have the url in their index.
You can put a no-follow meta tag in a page or a no-index. A no-follow meta tag, will allow Search Engines to crawl the page, but will not give link juice to any pages you have linked from that page, but as I stated, you will not keep the link juice, it will just evaporate. A no index will stop SE’s from indexing that page

What keywords are getting the high bounce rate?

Looking at your home page, its the sort of site after reading, you are either intrested or your not, unlike selling products where people may have a look at a few different pictures or products before making up their minds. I guess that does not explain a 90+ rate though

Has it always been so high, or is this a new thing?