From the data Benis has tracked and some additional data I've tracked, I haven't seen any evidence that the higher you are on the board, the more likely you are to go pro (with the exception that if you are near the bottom of the board and 60 guys higher than you get drafted, you don't go. In other words, there are no EEs who don't get drafted). It is possible that there is some slight difference, but if so, it is really small compared to what category you are in.
I have also seen no evidence that likelihood of going within a category varies by class. As shoe3 points out, what class you are in affects the likelihood that you fall into a particular category, but once you are in that category, your likelihood remains about the same, with likely going at about 87-90%, on the fence at about 40-45%, likely staying at about 10-13%, and staying at or near 0%.
The category a player is in can change throughout the course of a season. For juniors, if he is in the top 25 or so on the Big Board, you he is likely to be "likely going"; otherwise, he is likely to be "on the fence." For sophomores, if he is about a lottery pick, he is likely to be "likely going"; in the top 25 or so but not a lottery pick, he is likely to be "on the fence"; otherwise, he is likely to be "likely staying." I haven't seen enough very highly ranked freshmen to be sure, but I think they are the same as sophomores, except that if they aren't approximately top 25, they are "staying" instead of "likely staying." Note, that, if a player starts the season the big board in one of those positions, and then moves down the list, they appear to be more likely to keep that designation for awhile as they fall, and the opposite applies to those moving up (e.g., a junior player who starts the season as the 18th ranked player and drops to 27th by the end is more likely to be "likely going" than one who starts at 38 and moves up to 27).
The Big Board itself changes daily throughout the season, and continues to change all the way through the end of the NT. However, the final Big Board does not represent the exact order in which players are going to be "drafted" (I will call this the "Draft Order"). Only undergrads among the first top 60 players available to be drafted in any given year based on the Draft Order will declare. Obviously, all graduating seniors are treated as declaring, but, when deciding EEs, players only have the option of declaring if there aren't 60 players above them in the Draft Order who are either graduating or declaring. In other words, if the 100th ranked player on the Draft Order is "on the fence", his likelihood of declaring is 0% if there are 60 players higher than him who either graduate or declare. If there are fewer than 60, then his likelihood is about 40-45%. It's when there are fewer than 60 such players who are on the Big Board that allows for "off the board" EEs to occur.
Finally, the final "Big Board" does not represent the official Draft Order (unlike, for instance, the Projection Report). I believe, though I am not certain, that the primary difference between the final Big Board and the Draft Order is that the Draft Order incorporates any changes in the prestige of the teams associated with the players eligible for the draft, whereas the final Big Board only incorporates any within season game results (not overall prestige change). However, I also think it might be possible that there is either some small difference in the official Draft Order rankings vs. Big Board rankings, or some randomness associated with the order. I'm not sure though.