In observational studies with delayed entry, causal inference for time-to-event outcomes can be challenging. In addition to the potential confounding bias, the data can also suffer from selection bias due to left truncation, as well as bias from informative right censoring. To estimate treatment effects on time-to-event outcomes, inverse probability weighting (IPW) is often employed. However, IPW can be inefficient and is sensitive to model misspecifications. To address these challenges, we extend our earlier works and develop an orthogonal and a doubly robust framework for handling covariate dependent left truncation and right censoring. The framework can be applied to a variety of problems, including estimation of the average treatment effect (ATE) and the conditional average treatment effect (CATE). For ATE, we establish model double robustness and rate double robustness with respect to all three sources of bias: confounding, covariate dependent left truncation and right censoring. For CATE, we show that the orthogonal and the doubly robust learners can achieve the oracle rate of convergence. We apply the proposed methods to analyzing the effect of midlife alcohol consumption on late-life cognitive impairment, using data from the Honolulu Asia Aging Study.