1) Using the defaults for MatchIt, nearest neighbor matching matches on the propensity score as defined by a logistic regression of treatment on the covariates included in your formula. For each treated unit, it finds the one unmatched control with the closest propensity score, and then throws out the unmatched control units. There is no issue with continuous vs. categorical covariates here. Note King & Nielsen (2016), who describe why propensity score matching can actually make balance worse, as in your example.
2) MatchIt creates matches for the ATET, but the Matching package, which also implements genetic matching, allows you to specify that you want the ATE. After matching, you can simply perform the regression analysis you would have had you randomly assigned your units (assuming balance has been achieved).
The documentation for Matching is sadly fairly incomplete, leaving what it does quite mysterious. What is clear is that it takes a different approach from Stuart (2010) (and the Ho, Imai, King, and Stuart camp) in estimating treatment effects and their standard errors. Rather, it takes heavy inspiration from Abadie & Imbens (2006, 2011), who describe variance estimators and bias-correction for matching estimators. While Stuart and colleagues consider matching a nonparametric pre-processing method that doesn't change the variance of the effect estimates, Abadie, Imbens, and Sekhon are careful to consider the variability in the effect estimate induced by the matching. Thus, the analysis that Matching performs is not described in Stuart (2010).
The philosophy of matching described by Ho, Imai, King, & Stuart (2007) (the authors of the MatchIt package) is that the analysis that would have been performed without matching should be that performed after matching, and the benefit of matching is robustness to misspecification of the functional form of the model used. The most basic model is none at all, i.e., the difference in treatment group means, but regression models on the treatment and covariates work too. This group argues that no adjustment to the standard error is required, so the standard error you get from the standard analysis on the matched sample is sufficient. This is why you can simply export the matched sample from the output of MatchIt and run a regression on it, forgetting that the matched sample came from a matching procedure. Austin has additionally argued that standard errors should account for the paired nature of the data, though the MatchIt camp argue that matching doesn't imply pairing and an unpaired standard error is sufficient. Using cluster-robust standard errors with pair membership as the cluster should accomplish this. This can be done using the sandwich package after estimating the effect using glm() or by using the jtools package.
The philosophy of matching used by Matching considers the act of matching to be part of the analysis, and the variability it induces in the effect estimate must be taken account of. Much of the theory used in Matching comes from a series of papers written by Abadie and Imbens, who discuss the bias and variance of matching estimators. Although the documentation for Matching is not very descriptive, the Stata function teffects nnmatch is almost identical and uses all the same theory, and its documentation is very descriptive. The effect estimator is that described by Abadie & Imbens (2006); it's not a simple difference in means estimator because of the possibility of ties, k:1 matching, and matching with replacement. Its standard error is described in the paper. There is an option to perform bias correction, which uses a technique described by Abadie & Imbens (2011). This is not the same as performing regression on the matched set. Rather than using matching to provide robustness to a regression estimator, the bias-corrected matching estimator provides robustness to a matching estimator by using parametric bias-correction using the covariates.
The only difference between genetic matching and standard "nearest neighbor" matching is the distance metric used to decide whether two units are near to each other. In teffects nnmatch in Stata and Match() in Matching, the default is the Mahalanobis distance. The innovation of genetic matching is that the distance matrix is continuously reweighted until good balance is found instead of just using the default distance matrix, so the theory for the matching estimators still applies.
I think a clear way to write your methods section might be something like
Matching was performed using a genetic matching algorithm (Diamond &
Sekhon, 2013) as implemented in the Matching package (Sekhon, 2011).
Treatment effects were estimated using the Match function in
Matching, which implements the matching estimators and standard error estimators described by Abadie and Imbens (2006). To improve
robustness, we performed bias correction on all continuous covariates
as described by Abadie and Imbens (2011) and implemented using the
BiasAdjust option in the Match function.
This makes your analysis reproducible and curious readers can investigate the literature for themselves (although Matching is almost an industry standard and already well trusted).
Abadie, A., & Imbens, G. W. (2006). Large Sample Properties of Matching Estimators for Average Treatment Effects. Econometrica, 74(1), 235–267. https://doi.org/10.1111/j.1468-0262.2006.00655.x
Abadie, A., & Imbens, G. W. (2011). Bias-Corrected Matching Estimators for Average Treatment Effects. Journal of Business & Economic Statistics, 29(1), 1–11. https://doi.org/10.1198/jbes.2009.07333
Diamond, A., & Sekhon, J. S. (2013). Genetic matching for estimating causal effects: A general multivariate matching method for achieving balance in observational studies. Review of Economics and Statistics, 95(3), 932–945.
Ho, D. E., Imai, K., King, G., & Stuart, E. A. (2007). Matching as Nonparametric Preprocessing for Reducing Model Dependence in Parametric Causal Inference. Political Analysis, 15(3), 199–236. https://doi.org/10.1093/pan/mpl013
Stuart, E. A. (2010). Matching Methods for Causal Inference: A Review and a Look Forward. Statistical Science, 25(1), 1–21. https://doi.org/10.1214/09-STS313
Best Answer
If you use glm estimation that uses all the relevant covariates, you don't "have to" use matching at all, let alone include the weights in the model estimation.
But if you do, you can make your results more robust against possible misspecification of the model (your glm model might not be perfectly adequate). This is the whole point of the paper:
Furthermore, including the weights from the matching into the regression can reduce the variance and bias of the estimate.