Does DNA analysis allow determining amount of chromosomes?
This depends completely on the quality of the DNA. Since each chromosome is essentially a very long strand of DNA, breakages and missing sections are very common in extinct species due to degradation over time. 

If a full DNA read is absent, no determination of chromosomal number can be performed. 

Assuming a full read (covering all breakages) of the DNA is present, the telomeres of each chromosome can be found, and since chromosomes usually only have one set of telomeres each, this may be useful for chromosomal number determination. 

Caveat 1: Chromosome 2 (human) is an example of how this method may fail, since a recently fused chromosome will still possess the tandem repeats characteristic of telomeres. 

Caveat 2: Telomeres may vary in length due to the age of the cell, as well as be broken in multiple regions. Due to the tandem repeat nature of telomeres, it is nearly impossible to tell these different states apart experimentally.