Number of transcription factor genes in the human genome
Here I will assume we are talking about eukaryotic sequence specific transcription factors (ssTFs) and try to answer your first and part of the second question. There is in any case not definitive answer yet.

An estimate of ssTFs genes in humans is given in the 2009 Nature Reviews Genetics paper by Vaquerizas, JM et al, A census of human transcription factors: function, expression and evolution.

An excerpt from the abstract:


  Here, we present an analysis of 1,391 manually curated sequence-specific DNA-binding transcription factors, their functions, genomic organization and evolutionary conservation.


The numbers are somewhat higher now. Wingender et al. have counted 1,558 human genes in their TFClass database 2013 NAR paper. In their 2014 NAR paper they included 1,557 human, 1,147 mouse and 1,105 rat orthologs.

Another way to look for this information is to see the number of entries listed in TF databases, like for example JASPAR. This has the advantage of including other species. However, the coverage here depends on the availability of position weight matrices (PWMs) for the binding specificities. Many uncharacterized TFs may not be found.



To try to answer your third question, that is, what is the proportion of TFs in the different species, a naive approach would be to divide the number of predicted TFs by the number of predicted genes in the target genome. For example, taking the latest estimates above with the predicted number of coding genes from Ensembl database (version 78) will return these percentages:

# Human
100 * 1557 / 20364 = 7.64%
# Mouse
100 * 1147 / 22606 = 5.07%
# Rat
100 * 1105 / 22777 = 4.85%


This suggests humans have a slightly higher proportion of TFs than rodents. However, these differences are not too big and may be dependent on the accuracy of the different estimates on TFs and gene numbers. An in itself, these numbers are not so interesting.

A much more interesting question is whether TF families have expanded more or less in different species (that is, whether the number of  proteins within each family has increased, regardless of the proportion to the total number of genes in the genome). I could find at least one paper where this has been done systematically for several eukaryotic species, covering animals, plants and fungi, and focusing on TFs common to the ones found in plants. The main conclusion of the paper is that some TFs families have expanded more in plants than in other organisms. Quoted from the abstract:


  To investigate if differences exist in the expansion patterns of TF gene families between plants and other eukaryotes, we first used Arabidopsis (Arabidopsis thaliana) TFs to identify TF DNA-binding domains. These DNA-binding domains were then used to identify related sequences in 25 other eukaryotic genomes. Interestingly, among 19 families that are shared between animals and plants, more than 14 are larger in plants than in animals. After examining the lineage-specific expansion of TF families in two plants, eight animals, and two fungi, we found that TF families shared among these organisms have undergone much more dramatic expansion in plants than in other eukaryotes. Moreover, this elevated expansion rate of plant TF is not simply due to higher duplication rates of plant genomes but also to a higher degree of expansion compared to other plant genes.