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Greedy colouring visits vertices in decreasing degree order and gives each the smallest colour not used by a neighbour. Brooks' theorem guarantees at most Δ colours suffice unless the graph is a complete graph or an odd cycle.
Greedy bound
χ(G) ≤ Δ + 1, and χ(G) ≤ Δ unless G is complete or an odd cycle
Greedy uses 3, matching the triangle 1-2-3 which forces at least 3.
No. It is optimal here but can be off for adversarial vertex orderings; exact chromatic number is NP-hard.