We develop new inference procedures for a linear IV model that are robust to identification strength and heteroskedasticity of unknown form, and nonparametric with respect to the first-stage equation. Our first test is tailored for inference on parameters of endogenous explanatory variables. Our new statistic modifies that of Antoine and Lavergne (2003) to directly account for heteroskedasticity of unknown form. As a result, it is asymptotically pivotal, so that inference is greatly facilitated in practice. We also develop (i) an identification-robust subvector inference procedure that does not rely on the knowledge of identification strength for the remaining parameters, and (ii) a pure specification test. In both cases, the tests are conservative but powerful. We show that our procedures are computationally friendly and competitive with existing ones in simulations and an application.