To answer your main question first:

No, because Likert data is not numeric. The values are categories. You can name them, but there is no quantity associated. The only additional information we have is about the order of the categories/names.

By assigning numeric values to the Likert values you create a (discrete) random variable. This is not principally different to any other (discrete) random variable. There is a function that assigns probability values to possible outcomes/events. The problem here is that we usually have not much an idea about how this function looks like. The distribution can be anything, and it is not granted that it is unimodal, symmetric or expressable by some simple functional form. That makes it difficult to guess if assumptions needed for data analysis are sensible.

The meaning of the (numerical) values is outside of any statistical consideration. The researcher must justify why the assignments of the chosen numeric values makes sense, and how calculated quantities (sums, differences, means) may make sense. If a reasercher cannot give a plausible argument why such values would make sense (i.e. what they actually quantify), it would be better to not go beyond an ordered logistic regression analysis (where random variable is constructed to be made of zeros and ones only, so that sums reflect counts [of occurences], means reflect proportions [of occurences], and the analysis is about odds ratios).

So, when a reasercher is talking about a "mean of Likert values" he actually means the mean of the constructed random variable, and as long as the mapping from the Likert values to the numerical values is not sensible, the given quantities are not sensible either.

1) Yes, Likert data is not related to counts. If it was, one could either count directly or measure, physically, some attribute that is related to the quantity (like we measure a mass or weight rather than counting the molecules).

2a) No, because Likert data is not numeric (it is a fundamental assumption of all methods based on parobability theroy that the thing under analysis is a random variable: a function returning numerical values!). However, the constructed random variable can be analyzed as if it was a continuous random variable. If it makes sense, depends on the actual properties (number of different categories), the meaningfulness of the value mapping, and the existence of a functional form that is useful to describe the probability distribution.

2b) No. NegBin is a model for count data. Sa counts cannot be fractional, this model can not handle frational values.

If the random variable that is constructed by mapping numerical values to Likert values can resonably be described by a continuous model, then I think the most obvious choice is a beta-distribution. However, this might not be sensible in every case!

If the random variable is discrete and only takes consecutive integer numbers, a binomial model would the most obvious choice, IMHO. Again, if this makes sense is to be decided in each case.

Dear Tim,

   I will respectfully disagree with JW on a few points.

   All models are false but some are useful. While the motivation for negbin arises from counts, the parametrization actually describes a probability over a set of integer values. These occasionally can be _described_ by a negbin distribution. If the data were well described by a distribution we might, sometimes, even be able to proceed to inferences. All without ever thinking of likert values as reflection counts of anything.

   The question, then would be what are you actually testing in this case? Because to the extent that the underlying judgements really are ordinal then it is hard to explain what your hypothesis test actually reflects.

2. Can you deal with it as a continuous variable? This has a long and dishonourable history motivated by the argument that the "binning" of a continuous variable into 7 points has "limited" damage on our estimates of the mean and variance. In practice this bad idea will only get worse because anything but a uniform discrete distribution in your results will mean that effectively you have even fewer than 7 categories to play with. 

   One also needs not to forget that the fundamental ordinal judgement that leads to these ratings guarantees that you will misrepresent your findings by pretending that they are an unbiased physical measure....

2b) Using negbin for continuous data will not often make sense. In addition to the Beta distributions, the Weibull can also be useful for data with the sorts of shape that would cause you to think it was relevant in the first place. 

Good luck

The above answers are very formal and theoretical. On a much more pragmatic level, many researchers do treat a Likert type variable (coded say from 1 - 7) as if they were continuous variables, and get sensible answers out. Since your Likert variable is probably skewed, I would be inclined to at least try quantile regression, and see whether the results make sense.

My colleagues above would probably tell me I am crazy for this advice!