About assumptions needed to get convergence of stochastic gradient methods on non-convex objectives
There is a lot of recent work on these questions spurred by interest in deep learning and other non-convex optimization tasks. If the objective is differentiable and smooth (i.e. if the gradient is Lipschitz), then you do not need to assume bounded gradients and there are quite a few assumptions on the noise you can adopt, and they are reviewed (with convergence results) in Khaled and Richtárik (2020) (conflict of interest: I'm a coauthor of this paper). 

If the objective is not smooth, then all the analysis of SGD I know of assumes bounded subgradients possibly in addition to other regularity properties (see Davis and Drusvyatskiy (2019) and Zhang et al. (2020), although the latter does not analyze the vanilla SGD per-se). Note that if you enforce some sort of iterate normalization mechanism, feedforward ReLU networks do have bounded subgradients. 

Unfortunately, even neural networks with differentiable losses are not globally smooth. For Adam and AdaGrad, it turns out that you do not need global smoothness as Défossez et al. (2020) show. There are a lot more results specifically for neural networks, but I will refrain from mentioning them since your question was about general assumptions.