You can include variables that were used in the matching in the outcome regression model used to estimate the treatment effect. Ho, Imai, King & Stuart (2007) argue that you should perform in the matched dataset whatever analysis you would have performed in the original dataset, now with some added robustness to model misspecification. If you exactly matched on any of these variables, including the variables in the outcome regression will not change the effect estimate or standard error. It's a good idea to include covariates in the outcome model after using coarsened exact matching (especially continuous variables or collapsed categorical variables) because the matching will not completely eliminate the imbalance in these variables.
You do not have to match on all variables that affect the outcome to arrive at an unbiased or low-error estimate of the treatment effect. To address confounding, you need to adjust for (either through matching or regression) a sufficient set of variables, which can be determined using graphical criteria. See Elwert (2013) for a nice introduction. I also explain confounders in this post. As long as your conditioning strategies together adjust for the required variables (and you have got the set of required variables correct), you can have some confidence in the validty of your effect estimate as unbiased. You can match on some variables and then use regression on the others. This is generally not a recommended practice, though; you should attempt to use all methods available to adjust for variables you need to adjust for. Sometimes, however, you can incidentally achieve balance on some variables when matching on others. Some research has shown that there isn't much value in including covariates in a regression when the standardized mean difference is below 0.1 (Nguyen et al., 2017), though I wouldn't take that result too seriously.