Is it possible for cells from the same tumor to have different genetic material, and if so, to what degree is it possible (how fast do they mutate) ?
Cancer cells and normal cells differ on the genetic basis but they share the same genetic background, so they have not different DNA in the sense of two different people. They have to be different, since cancer cells have to accumulate mutations on a number of genes to become a cancer cell, which can survive and will not be directed into apoptosis. These are genes which control the cell cycle, certain growth factors, tumor suppressor genes, cell signalling and so on. In all these genes there need to be either activating or deactivating mutations present. Most of these mutations are point mutations (single nucleotide polymorphisms, SNP), where only one base is altered to achieve a mutation. An example for such a mutation would be the mutation in the B-RAF kinase, which is involved in signaling and activating genes, where a point mutation exchanges Valine 600 to glutamic acid (V600E, for reference see here). These are relatively small differences between these two cells. The Cancer Genome Project aims to sequence cells from a cancer and also normal cells from the same individual and then do a comparison. Later stages tumors often tend to genetic instabilities where complete regions of the genome are duplicated, inverted or deleted, see here for a review. A more basic overview is given by the Ebook "Essentials of Cell Biology" by the Nature Group, which has an overview of cancer cells.
Regarding the mutation rate, this is hard to estimate. There are a few papers available here, but they focus on the mutation rate in germ cells, so these are mutations that are transmitted to the next generation. The estimates there are between 70 and 100 mutations per generation per individual, depending on the research method. You can find a nicely explained blog article here, which gives a number of original references, too.
The mutation rate in cancer is a different thing, since this depends on which genes are mutated. The mutation rate for cancer has to be higher for the cancer cells to allow to collect the necessary transformations into a cancer cell, but these mutations seem to occur only in a few hotspot regions of the genome (Further references can be found in this article called "The causes and consequences of genetic heterogeneity in cancer evolution" and in this called "Emerging patterns of somatic mutations in cancer"). There are two article called "The mutation rate and cancer" which are interesting in this context. They can be found here and here. As always, if you have problems with getting the articles, let me know, I can help there.
A) It absolutely is very common for extensive genetic heterogeneity to exist in tumours, including between individual cells. Talha already posted links to some very good reviews from Charlie Swanton and colleagues so I will just add in a citation for single cell evidence here ( http://www.nature.com/onc/journal/v36/n20/full/onc2016438a.html )
B) As for mutation rates - they tend to markedly vary across cancers and across regions of the genome based on replication timing and chromatin boundaries. Given the variation in the total number of mutations accumulated and assuming that a similar number of cell divisions , given birth and death rates, are required to get to a typical clinically observable tumour there must be variation in the underlying mutation rate.
Such variability may have much to do not only with background rates of mutagenesis but also with deficient repair mechanisms coupled to it, and there are clear examples of aberrant repair contributing markedly to mutational burden, including mismatch repair ( http://www.nature.com/onc/journal/v36/n20/full/onc2016438a.html ) and homologous repair defects ( https://www.nature.com/ng/journal/vaop/ncurrent/full/ng.3934.html ) and mutational processes (UV-induced and smoking induced cancers have particularly high mutational burdens)