Many non-savvy internet users may not even be aware of the existence of other extensions. That said, because there are many new extensions popping up every now and then, you can often find amazing one word and 3-letter domain names with these new extensions that are otherwise almost impossible to find unregistered in the .com world. Additionally, these new extensions are becoming more and more familiar to users every day, hence it's almost certain that their value will continue to increase as time passes and wider adoption manifests. 
Length: This is pretty self-explanatory. Generally the shorter the domain name, the more valuable it is. This difference in value decreases exponentially as the name gets longer, though. For instance, the difference in price between 3 and 4 letter domains are significant, while the price difference between a 7 and 8 letter domain would usually be much less significant.
If the third party in question does not have the trademark rights they claim, or it is not a very strong claim. In most cases, the third party will need their trademark registered with the US Patent and Trademark Office (USPTO). The USPTO website has a trademark search engine that you can consult. This may also require a cybersquatting attorney to help determine if the trademark claim is strong enough.
Perspective buyers are able to directly contact domain holders in cases where the desired domain is no longer available. Most registries openly publish the names and contact data of domain holders. Once this information is gained, buyers can get in touch with domain holders and make an offer for the name. Sales are also known to occur in instances where the original domain owner had no prior commercial ambitions
But in general, the best and easy way to sell a domain is to go with a marketplace. There is many popular marketplace dedicated to selling domains including Flippa and SellDom to sell your domain (full disclosure, I’m working on SellDom). The advantage is to leverage the popularity of that kinds of services in hope that someone will find your domain and buy/make an offer.

Since search engines regard inbound links as recommendations, these links have a decisive influence on the ranking. The search engine definitely doesn’t appreciate link networks and won’t give new projects a jumpstart by allowing a former website’s backlink profile to be used when registering a new domain. Using an expired domain is like playing with fire, where the worst case scenario is that you’ll be penalized or be excluded from the search index.

So how do you develop that instinct? NameBio maintains a database of over 500,000 historical domain sales (as of writing this post). They have interesting filtering features by which you can narrow down domains by price range, date sold, keywords and more. Simply sifting through the listings on NameBio long enough will quickly develop your domain appraisal “instinct”.