I know that point mutations can change the base sequence of a gene by altering a specific codon that codes for a particular amino acid. Are these mutations purely random events that occur when DNA is being copied into a cell? Furthermore, can environmental factors play a role in the amount of mutations that occur?
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can environmental factors play a role in the amount of mutations that occur?
Yes! The environmental factors that increase the mutation rate are called mutagens. You will definitely want to have a look in the wikipedia article. You may also want to read the wikipedia page on mutagenesis
Some chemicals increase the mutation rate, physical agents such as radiation also increase the mutation rate. There are viruses that are mutagens as well. Also, the number of mutations that an organism produce can also be phenotypically plastic. For example, plants have been shown to increase thee number of mutations they produced in response to stressing environment. The reason (or at least the commonly stated reason) is that they produce many mutations, "hoping" that at least one of the offsprings will carry a beneficial mutation and will be very fit even in this environment that is stressing for the parents. Many viruses and some bacteria also act as mutagens.
Are these mutations purely random events that occur when DNA is being copied into a cell?
Mutations are random in the sense that it is impossible to predict with certitude that a given mutation will occur at a given locus (locus=position in the genome). As a consequence, it is also true that mutations are random in the sense that it is impossible to predict the effect on the phenotype (and on the fitness) of a future mutation. Mutation rate varies throughout the genome. Typically, repeated DNA sequence tend to have higher mutation rate. Also, transposons causes many mutations and regions in the vicinity of transposons have higher mutation rate. But again, you should have a close look to the wikipedia pages I linked above.
Note that a full answer would require going into the terminology of the term "random" which itself is a question of philosophy. When using the term random, it is always important to specify what is meant exactly (which is not done in the question) as one may argue that since the Big Bang happened everything has been deterministic.
There are many different types ('scales', you might say, and mechanisms) of mutation in the genome.
Maybe your question was really about base mutations, but I want to add one interesting tangential thing (which also points out the importance of clarifying the term 'mutation'):
Segmental Duplications - non-random genomic mutation
There are mutations that you would not call random; that occur again and again in patients of some disorders. By studying patients with Charcot-Marie-Tooth 1A (and other diseases), geneticists identified structures in the genome which act as 'platforms' for large-scale mutations i.e. chromosomal rearrangements.
Certain places in the genome are rich in segmental duplications, paralogous segments of the genome which can be reasonably large (1-400kb), may contain genes, and have very high identity (>90%) making them prone to recombine.
The presence of segmental duplications (SDs) creates local genomic instability; a propensity for recombination, causing rearrangements - the term "genomic disorder" was coined for diseases arising from these rearrangements.
These genomic disorders are typically gene dosage imbalances. In the case of Charcot-Marie-Tooth 1A, duplication of a region between 2 SDs contains a gene involved in myelination. Deletion of that region causes another neuropathy. Haemophilia A can be caused by the inversion of a region between 2 SDs which lies partially within a long gene encoding a clotting factor. Inversion totally disrupts the exons.
Read more here: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1226196/