Since you'll be in Switzerland on a long-term visa (and therefore a resident), nothing would stop you from applying for a Schengen visa from within Europe. There are plenty of European consulates in Bern and other major cities, however most of them don't issue short-term visas. The reason is that Switzerland is a Schengen member and very few local residents require a visa. E.g. from the Czech embassy in Bern:
Due to the possibility to travel in all countries of the Schengen-area
using only one visa the Ministry of Foreign Affairs has decided to
unite the Embassy Visa Departments into Regional Visa Centers.
As of January 1st 2011, therefore, visa-applications are only being
processed in the Consular Departments of a few selected Czech
Embassies within the Schengen-area (Athens, Berlin, Bratislava,
Brussels, Madrid, Stockholm, Warsaw, Vienna). Their territorial
competence is the following:
Embassy of the Czech Republic Vienna (Austria, Italy, Switzerland, Malta, Slovenia)
So, e.g., in order to obtain a Czech short-term visa, you will need to travel to Vienna. As statistics found by @Relaxed show, only 162 visas were issued within Switzerland in 2014, so it's unlikely you will be able to find a consulate to help you out within the country. To answer your other questions...
Am I able to apply to extend or renew my Schengen visa while I'm already in the area
You have a national 'D' visa, which might indeed be extendable if you have a reason to stay in Switzerland for more time. However you cannot 'convert' a 'D' visa into a tourist ('C') visa, so you would need to apply for one from scratch.
Would it be possible for me to go to UK and apply for the Schengen visa there?
No, since you won't be a UK resident. See our related question for more details.