Hi!
In reply to your questions.
1. No it would be impossible for a airline the size of Delta to be able to track web users searches unless a user signed in (such as occurs in Expedia for example) with their Frequent flyer info (as in Emirates.com) and even this is remote possibility given the number of users who use the website on a daily basis.
2. There is a way to know the fare, when booking online as i just did on the DL website in advanced search it allows you to select the fare level starting with E and going up all the way to Y in economy and again when you select your flights and want to confirm as you see on the screenshot you can see the fare class that was booked.
3. The fare level that is showing is the most current available fare and that does not mean that it cannot fluctuate as bookings get cancelled and more cheaper options become available or wont go up as the cheaper seats become full.
4. Yes, generally booking well in advance of the date of travel will assure you of a good price but this is always dependant on other factors such as availability.
Hope this has helped.
rgpnz.