At what point in the cell cycle do cells start to become tumorous? Do they have abnormal characteristics to begin with; if so what are they?
Cancer cells don't start to become cancerous at a specific stage of the cell cycle; you will find that while uncontrolled proliferation is a hallmark of cancer, different cancers acquire alterations in different phases of the cell cycle. BRCA-deficient cancers for example have a compromised G2-M checkpoint , while Rb deficient cancers have a compromised G1-S  checkpoint in the cell cycle.
The cell cycle is simply a property of proliferating cells and the same broad phases of the cell cycle are universal to both normal and malignant cells.
As for when abnormally growing cells actually become a cancer - this has nothing to do with phases of the cell cycle, and everything to do with the ability to break through the basement membrane of the original site (indicating the potential to be invasive/spread), because the ability to invade is a hallmark of cancer 
The question assumes that tumorigenesis is a cellular problem. My answer is a counter-question: Is tumorigenesis a cellular problem?
Sonnenschein & Soto (2018) Methods Mol Biol 1702: 15-26; emphasis mine:
[T]he tissue organization field theory (TOFT) (...) is based on the premises that cancer is a tissue-based disease and that proliferation and motility is the default state of all cells.
Sonnenschein & Soto (2016) Prog Biophys Mol Biol 122: 70-76; emphasis mine:
For a century, the somatic mutation theory (SMT) has been the prevalent theory to explain carcinogenesis. According to the SMT, cancer is a cellular problem, and thus, the level of organization where it should be studied is the cellular level. Additionally, the SMT proposes that cancer is a problem of the control of cell proliferation and assumes that proliferative quiescence is the default state of cells in metazoa. In 1999, a competing theory, the tissue organization field theory (TOFT), was proposed. In contraposition to the SMT, the TOFT posits that cancer is a tissue-based disease whereby carcinogens (directly) and mutations in the germ-line (indirectly) alter the normal interactions between the diverse components of an organ, such as the stroma and its adjacent epithelium.
In some ways this is simply a function of how we define 'tumorous'. We can trace the phylogeny of tumors in the same way we trace that of species, using DNA sequencing. In some tumors one sees a clear causative mutation at the root of the tree, so for these we might say that there was a single mutation which caused cells to become tumorous. Mostly though things are not so clear.
For instance, it's now believed that in many cancers, the rapidly dividing cells are descended from more slowly dividing 'cancer stem cells'. These cell cancer stem cells tend to survive therapies that target more rapidly dividing cells, such as radiotherapy or many chemotherapies. In such cancers we could talk about a mutation by mutation transition from non tumorous to tumorous cells.