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I wouldn't get to concerned about having an a file extension in your URL. If you can remove/hide it, by all means go for it because it couldn't hurt + it looks a little nicer.  With that said though, I wouldn't go out of my way to hire a developer to do that, it's not going to make a major impact on your ranking ability.

I wouldn't go out of my way to build a bunch of profile backlinks, but I would definitely get profiles made up on sites that make sense to use for your industry.

On a side note though, services like KnowEm can be useful for reputation management. Even if those profiles don't rank or pass any value along to your actual site, you can (and should) prevent someone from using your brand name.

I would probably go with the PDF, possibly with a thumbnail JPG image of it instead of a text link to the PDF. That's just me personally.

Most RSS feeds exist because a site has some type of content generating script/software tied to it (IE a blog.) If your site doesn't have something like that already, you can create an RSS feed using a site like this: http://page2rss.com/

If you are unsure if your site has a feed, in Chrome, you can use this extension to find out - https://chrome.google.com/webstore/detail/pnjaodmkngahhkoihejjehlcdlnohgmp

Any link that points to another page could be considered a "backlink" - these doesn't mean every link will show up in "back link reports" from various places.  Every data set is different.

For example: In Google Webmaster Tools, it shows me a handful of links for my site. Bing Webmaster Tools happens to show the same links, plus some that Google didn't. The same can be said about all link tools (Open Site Explorer, Majestic, Ahref's, etc)

You can acquire links through an uncountable number of ways. The type you are currently chasing though may not be the best route for long term strategy or success.  I'd suggest checking out link building related posts here on SEOMoz and then elsewhere.

If you just want to jump into action without really getting a better idea, this site has an awesome list of link building ideas. However, keep in mind there is a certain skill level involved with how you go about building links, something you can't fake easily.

This is personalization. Nothing completely new, it's just getting more sophisticated. Your SERPs will vary based on if you are logged into a Google account, your browser history, your IP address's location, etc.

Enter private browsing mode (Firefox) or Incognito mode (Chrome) for a better chance of getting less personalized results.  You can also "verify" your rankings by using multiple rank checkers.

Chances are you won't get the exact same answer across them, but you'll get a much better idea than manually checking it with all the "personalization variables"

As John Pring put it, OSE doesn't update instantly or "often" - you might want to try out Google Webmaster Tools and Bing Webmaster Tools.

Bing's inbound link tool is remarkable. It's showing links to my site that I've never seen before (and I've tried OSE, Google's, and a few other "back link reporting" tools)

I'm most familiar with myblogguest.com but I wouldn't discredit any of those sites. Take some time to look around all the sites to see if you can find people looking for content that is somehow related to your niche.

I typically take a look at whatever site I have plans to guest blog onto, and see what their topics are like. Then I'll look for any "information gaps" on their site that would fill a need (for them), as well as be a relevant topic to the site you're going to be linking to.

This way you're getting a nice relevant link, but more importantly, you're producing something that the blog actually needs and will** benefit** from.  That might get the guest post more exposure as well as strengthen your relationship with the owner of the blog.

If you are on an Apache server and have FTP access, you can setup 301 redirects using .htaccess

The proper format for this is as followed:

redirect 301 /oldurl.html http://domain.com/the-full-url

As Ben Fox stated, you can use the report to find the linking errors.

I'd also run a scan of your site using Xenu Link Sleuth (it's 100% free) if you're a PC user. Some people prefer Screaming Frog (both work well, Screaming Frog has a free and paid version to my knowledge)

I use Xenu personally, been using it for years with much success. You'd be surprised what kind of stuff it digs up.

Here's the format you may need to follow, I'm not 100% positive this will work for you but I know this is how 301s are setup:

redirect 301 vacancy.php?id=802  **yourdomain.com/path-to-location/**vacancy/?id=802

Totally forgot about that. CSS galleries are a great way to get a link.

Some ideas that may work for you...

Look for local websites of active bloggers. See if you can get a guest blogging position, or donate a brand new design for their site in exchange for a plug/review of your service from the blogger.

You can also look for local non-profit organizations that accept donations. Sponsoring/donating is a great way to get a link and help a cause (which may also lead to spreading brand awareness.)

You might also want to contact authority sites that are either local to your region or at least within the web design industry. Create some type of high quality content (infographics, video, free eBook, etc) and have the authority site host it.  It's a similar concept to guest blogging (giving something of value to someone else.)

Lastly, you can at the very least, join local Chambers/business networks and get listed on their membership page (if it exists. If it doesn't, why not recommend that they have one for all their members, you can even create it for them since that's part of your industry)

Lots of options, really depends how dedicated you are and how many resources you have available.

My guess is either developer preference or lack of knowledge. Joomla is the same way. You can have domain.com/about  or domain.com/about.html - both accomplish the same thing, one takes an extra step to do (adding the .HTML suffix.)

I guess if you are coming off of an old static HTML site and moving to a CMS, including the .html would allow you to keep the same URL structures much easier than setting up 100s of redirects.

Nope, either should be fine. The biggest thing to remember is if you change any URL structure, setup 301 redirects to go from the page's old URL to the new URL.

I definitely find commenting on blogs to be a great way to get involved and gain attention. The link you may get from it should be seen as a bonus to the actual engagement and brand-awareness you are gaining.  I developed a tool to help speed up how quickly you can find related blogs to your niche, as well as finding ones which allow custom anchor text (and sometimes are followed.)

You don't want to over do it, but being involved in your industry (on the web), is sometimes equally as important as being involved in the real World.

I would consider other routes than using article directories for a long term strategy. I'd suggest placing one quality article on each site (totally unique, not spun.) Then try to build relationships with bloggers in your niche and try out guest blogging. You'll get the benefit of the link, the anchor text, and also a dedicated audience base somehow related to your niche.  In short, the benefits are much greater.

If you must go the article route, definitely get links from a variety of sites. Make sure they aren't all ran by the same people. There's a lot of article "networks" which are typically easy to spot. It's beneficial to check the WHOIS data for the domains and the IP address/name servers being used by the domains to make sure you aren't willingly placing your articles on a "link network."

This is exactly how I handle similar things. Any badges or widgets should either be branded with just your domain URL, or brand name. Looks natural, also looks better for the end-user (the people visiting the website that the badge/widget is in)

Without seeing more details, as Steven Macdonald stated, I don't think anyone can really help you out.   When you look at your page vs the pages ranking above you, how do they vary in terms of keyword usage, page content, rich media, title tags, URL structure, etc? One item in the Moz "Page Grader" that can hurt your rankings is the page structure. If you adjusted that, search engines would have to re-rank your page.

If your page was domain.com/keyword and you change it to domain.com/keywords then you are technically ranking a new URL and should setup a 301 redirect from the old URL to the new one.

Without knowing what you really did, it would be hard to tell.  I would suggest going to the page that dropped and checking Google's cache of the page.

If Google's cache still shows your old content (pre-Moz suggested changes), then something else is the problem.

I would also be sure to check Webmaster Tools to ensure that you haven't been sent any type of messages / warning notifications.  That's a pretty big overnight drop to be purely content-related in general.