I know (and have read in other posts) that logistic regression isn't the only way to calculate propensity scores. But if you do want to use logistic regression for that, must you then check the linearity assumption? And since it's the relationship between the independent variables and the log odds that should be linear, you must first run your logistic regression to get the log odds and then for example plot the independent variables one by one against the log odds to check that assumption, right? Finally, when calculating propensity scores in this way, do you need to check for multicollinearity between the x-variables?
You don't need to check for any modeling assumptions when modeling propensity scores. The agreed-upon way to assess the quality of propensity scores is to examine the degree to which conditioning on them yields balance in the matched, weighted, or stratified sample. Having a correctly specified propensity score model will (in theory) improve your ability to achieve balance, so it makes sense to try and construct a model as well as you can, but you are not performing any inference on the coefficients in the propensity score model, so the assumptions that typically accompany inference in logistic regression do not apply. Multicollinearity does not matter for these models because their sole utility is in the estimated predicted values from the model, which do not depend on the multicollinearity of the predictors. Again, the assumptions you mentioned would be important if you were attempting to make inferences about the coefficient estimates from the logistic regression model, but that is not a part of propensity score analysis.