Difficult is totally subjective. It's probably the most resource intensive just because of how many different pages there are, but to me that's not what makes a problem difficult.
Also, I could see this ranking high for consumer-level problems, but there are way more difficult (both resource intensive AND hard to figure out) and interesting problems when you look at some of the business applications.
Pretty much every startup/company is suffering from not having enough people to do development time, but I have seen this before. As odd as it sounds, getting things done is NOT a priority at these places. Sometimes it's more about politics or positioning or business-ey things but it's not real. It's hard for people who like to be productive to understand, but it's true.
I agree with Tom to get the heck out. If you want to make a last ditch effort, find the highest ranking person who will talk to you and make it 100% clear that you have NOTHING to do. Then if it's still all talk, get out.
NUnit is a MUST. I wonder why you say unfortunately. I realize it's not the most popular thing on the HN block but if you're not proud of your codebase, it's never going to work.
Since you have NO automated tests, start this way:
1) Every single time someone fixes a bug or adds a new feature, they MUST have at least one automated test that "proves" this functionality is correct
2) In your "spare" time, you can also write automated tests to test the code execution paths covered in your manual tests that I assume you run
As for what to skip, at the beginning: nothing. Try testing -everything- you touch. You need all you can to get some real regression testing in. Eventually when you are at 50-70% code coverage you will be in a position to know what are the high-value things to cover and maybe pick and choose (although probably not).
I also think it's unlikely that the intern actually weighed his argument and decided against it, and would have been in a position to make it happen had his arguments convinced him. Most likely he was told which features would not be implemented and is spewing that out to customers.