Usually, would one lag the independent variable?
If you are dealing with panel time series you will almost always want to include lags (even alongside contemporaneous variable) to explicitly model short run dynamics. Ideally you would want to set up model where there is no autocorrelation in an error term as that can be viewed as an indication that your model is not properly dynamically specified.
You can use lags to deal with endogeneity, however in most cases people would use lags as an instrument rather than getting rid of contemporaneous variable. However, you could do that and estimate a 'reduced form' model with just a lag instead, although you should in such case probably make some justification for doing so as people will probably have questions about that. There are also some other models that might be suited more to panel time series data with endogeneity such as panel VAR models (e.g. see Canova and Ciccarelli 2013 for survey of these models). Generally speaking people in macro do not typically just use standard fixed/two-way effects models for panel time series data as such models are more suited to situations where N>T.
Claiming there is delayed effect is not sufficient justification to only include lagged variable. Even if contemporaneous variable will turn out not to be significant including one extra variable will have only marginal effect on power of the test whereas excluding it could lead to omitted variable bias. Moreover, if you work with lets say yearly data such explanation would be much less justifiable then if you work with lets say monthly data.
Usually, would one control for previous values of the dependent variable?
Yes, this is actually quite important to do if you believe lagged dependent variable does/should affect current values of dependent variable. However, in such case you need to make sure you get the short run dynamics right and your model won't have any autocorrelation in residuals, since getting this wrong will make the parameter estimates inconsistent and you won't be able to save the model just by using some autocorrelation consistent errors (see Verbeek's Guide To Modern Econometrics pp 126).