In this paper, we consider multi-agent learning via online gradient descent in a class of games called λ-cocoercive games, a fairly broad class of games that admits many Nash equilibria and that properly includes strongly monotone games. We characterize the finite-time last-iterate convergence rate for joint OGD learning on λ-cocoercive games; further, building on this result, we develop a fully adaptive OGD learning algorithm that does not require any knowledge of the problem parameter (e.g. cocoercive constant λ) and show, via a novel double-stopping time technique, that this adaptive algorithm achieves the same finite-time last-iterate convergence rate as its non-adaptive counterpart. Subsequently, we extend OGD learning to the noisy gradient feedback case and establish last-iterate convergence results–first qualitative almost sure convergence, then quantitative finite-time convergence rates– all under non-decreasing step-sizes. To the best of our knowledge, we provide the first set of results that fill in several gaps of the existing multi-agent online learning literature, where three aspects–finite-time convergence rates, non-decreasing step-sizes, and fully adaptive algorithms have not been previously unexplored.