Will stratified log-rank test work if there is no event in some small strata? Since the analysis is for a rare event, it is possible that some small strata have no event. Hard to predict. Just wondering if there will be any problem.
Cox PH model will have problem for sure, but I am not sure about stratified log rank test.
The purpose of "stratified" is to adjust for baseline sex and age. We are not interested in the sex and age effect.

Neither the stratified Cox PH model nor the stratified logrank test will have a problem.  Strata with no event will contribute no information, and the baseline hazard will not be estimable, but $\hat\beta$ and the logrank test are fine.
(This is assuming you mean 'stratified' in the sense of the stratified Cox model, not in the sense of getting separate estimates in each stratum)
