Who or what companies can I contact that work with fragrance
companies? I would like to test new mens cologne and fragrances that
are still in testing phases and offer feedback to them.  Fragrance
companies would send me samples and I would offer the feedback and
suggestions regarding the usage of their products. I've done some
Google searches to no avail. Thank you.

Most companies have employees who are professionally trained to
develop new fragrances. They also have marketing departments that do
market research based on specific criteria. Do you have any reason to
believe that any companies actually do business by sending samples of
fragrances to random people for feedback, or are you just hoping that
some do? Because I would doubt that any companies actually do what
you're asking for.

I am a certified "mystery shopper" and I frequently evaluate stores. 
I was hoping that some companies want "Average Joes" to test their
fragrances and write up evaluations on my experiences.

If this information will cost more, I am willing to up my pay. Thank you.

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Most companies have employees who are professionally trained to
develop new fragrances. They also have marketing departments that do
market research based on specific criteria. Do you have any reason to
believe that any companies actually do business by sending samples of
fragrances to random people for feedback, or are you just hoping that
some do? Because I would doubt that any companies actually do what
you're asking for.
 
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magneta-ga is correct in supposing that's not how fragrance companies
work. I run one and know how it does.
There are around 3,000 new fragrances launched in the world every year
- over 95% of them bomb in the first few years.
There has been a lot of work done by fragrance houses - and I don't
mean the retail brands - I mean the perfume manufacturing houses you
have probably never heard of, who fight for the business - IFF,
Firmenich, Givaudan, Quest and Symrise are among the main players.
They employ hundreds of skilled perfumers and "qualified" marketing
staff, and (despite launch budgets larger than the GDP of a small
developing country)they still mostly get it wrong.
Recruiting a few "average Joes" has been found not to work a long time
ago - it  simply does not correlate with sales success.
There are a few blogs around with "free" fragrance reviews, but even
the bloggers would mostly admit they're voicing their own opinions,
and fragrance companies have little interest in them.
One of the problems is that everyone thinks they're an expert, but has
no evidence to support this - in some companies fragrance selections
may be done by the marketing department, or even the boss's wife.
That's just "noise".
One interesting predictor I've come across is using "semi-expert"
panels to assess fragrances "blind", and those fragrances still being
worn daily and voluntarily after 3 weeks have a chance of selling
well. First impressions are not helpful.
One problem with this is that brand, bottle and box are themselves
often more important than the fragrance itself - and this testing
method excludes these factors. (It is typical for the box and bottle
to cost 5 or 6 x the "juice" cost.)
Most perfume knowledge is kept in-house for commercial reasons - you
won't find much published, it only leaks out when a fragrance house is
working with a client to sell them a new fragrance concept, and most
of that is cute marketing stories.
On top of that, the best perfumers see themselves as highly creative
artists "painting" fragrance compositions - they often believe their
work drives market taste, and have little interest in how it judges
them.

For an interesting perspective, have a look at the articles by
Chandler Burr, Perfume Critic for the NY Times: www.chandlerburr.com

Also  his articles: http://www.chandlerburr.com/newsite/content/articles.php