The simple answer to the first part of the question is yes, you need to convert NEFs to JPEGs most of the time to upload to "the web". There are a handful of sites out there that are aimed specifically at photographers and will let you upload RAW files (like NEFs, CR2s or DNGs) to work with in an interactive editor (you work with a JPEG or PNG on your screen, send instructions to the server, which edits the RAW file and sends back an updated JPEG/PNG). And, of course, you can save the NEF as "just a file" anywhere you can upload arbitrary files (like on a "cloud drive"), but anyone who wants to see the picture will need to download the file and have viewer software that understands NEF files. To view pictures directly in the browser, the picture needs to be in a format the browser understands, an that pretty much means PNG (which can be huge and slow to download) or JPEG (which allows you to trade perfect fidelity for file size and download speed). A high-quality JPEG is usually the best compromise between file size and picture quality.

You don't necessarily have to do the conversion manually. If you use the File→Web service... option in ViewNX 2 to upload pictures, the conversion to JPEG will be part of the process. If you're uploading from the browser, though, and you're not using a site that understands NEF (and, specifically, the NEFs produced by your camera) then you will have to convert beforehand.

Whenever you touch a picture in the Edit view in ViewNX2, the Save icon (lower-right in the controls pane) should go from greyed-out to active. You don't have to save each edit individually; clicking the icon once will save all of your unsaved edits from that editing session. And if you try to leave the edit view with unsaved edits, you will be presented with a Save Edits? confirmation dialog. (There is a Reset button to bail out of individual edits.)