Is it firmly established, that mutations are sufficient for cancer?
Acquiring mutations (more than one) are the necessary (and sufficient) steps on the way to cancer. The current opinion is that cells need "multiple hits" to fully transform into a cancer cell. It is also called the Knudson hypothesis, the original paper is here.

For example melanocytes (the pigment cells in the body) which acquire the activating BRAF V600E mutation (which is an important step in getting a melanoma cell, this mutation is found in about 70% of all melanoma) is not enough to transform the cell. Other mutations are necessary to do this.

Among the things important for cells to become malignant are to escape the apoptosis mechanisms of the cells and also the immune system. Both are linked to the mutation, inactivation or epigenetic silencing of genes, so I would say genes are the key to cancer.

For further readings these two articles should be interesting:


Are oncogenes sufficient to cause human cancer?
Significance of multiple mutations in cancer