I want a rough estimate of the percentage of nouns that belongs to each gender category
According to a footnote in this study (Warning: PDF file):
The distribution of the three genders in German is as follows. There are 4164 monomorphemic nouns listed in the CELEX database (Baayen, Piepenbrock, & Gulikers, 1995). Fifty-one of these have multiple genders (e.g., der See [the lake]vs. die See [the sea]). Of the remaining 4113 entries, 1758(42.74%) have masculine gender, 1567 (38.10%) have feminine gender, and 788 (19.16%) are neuter
which means you can roughly say that 40% are feminine, 40% are masculine and 20% are neuter
I have been self-studying German by reading the German press for the past 27 months. In the process, I'm writing out almost all unfamiliar German words, including nouns. As of May 10, 2016, there are 23.000 words in my German word list. Today I counted the nouns in my word list and looked into their gender distribution.
It turned out that my list contains around 10.375 nouns, of which about 44% are feminine, 36% are masculine, and 19% are neutral. 47 nouns can be used with any of two genders (e.g. der/das Kalkül or die/das Soja) and Joghurt can be used with either of the three genders: der/die/das Joghurt
According to the duden.de, %46: die; %34: der; %20: das Two genders existence is 1.3%.
Here are some statistics from Canoo.net
weibliche Nomen: ca. 68’000 = 40%
männliche Nomen: ca. 52’000 = 31%
sächliche Nomen: ca. 49’000 = 29%
An interesting addendum (or extension to all words) to idober's answer:
If you assume that none of these three categories is favoured, and you look at the distribution of articles used, you get a similar result:
Now this is very rough, because you only take into account "der,die,das" while "die" is used in more feminine inflections than "der". There are more inaccuracies, but the rough picture is a nice estimate, I believe.
However, I find it interesting that you get a result that is very close to the study based on CELEX data.