Regular multiple regression looks like Y = beta + beta(x) +...+ error but I don't know how to account for the matching in this equation.
In general, you don't need to account for matching in the outcome regression equation beyond just using the matched samples (which in most cases will be smaller than the samples you started with because you discarded individuals from the treatment and/or control groups for whom there was not an adequate match found).
If you had applied "matching with replacement", allowing for a child in Group B to be used as a match for more than one child in Group A, then you would have needed to account for this in the model, for example, by including a Group B member in the regression as many times as he/she was used for a match (or assigning frequency weights). But in your case, with as many Group B children as Group A children, it is not needed.
Secondly, it isn't clear if I should include age and gender in the regression because they were matching criteria, although many of the other posts say they should be included but not analyzed. What is the rule?
The main purpose of including variables in a causal-effect regression model is to eliminate confounding (estimated association between exposure and outcome being distorted/biased because exposure is also correlated with another risk factor that is also associated with the outcome).
Matching is also used to reduce or eliminate the impact of a confounding variable because by creating similar distributions of that variable in the two groups we essentially eliminate any differences in the association between the treatment/exposure variable and the outcome that are due to differences in the confounding variable.
Your end goal is to get an unbiased estimate of the effect of having a disability on test scores. If your matching did what it needed to do in terms of removing confounding, then including or not including those age and gender variables in the regression should not have an effect on the estimate of the disability variable. What is possible though is that if you are treating age as a continuous variable (rather than discrete, like gender), then there may be some interaction effects between disability and age that were not captured by matching. In that case, you could see a difference in results, but that would be due to the variables you are working with, not due to some inherent difference between controlling or not controlling for matching variables.
Also FYI, mathematically, if there are no other covariates besides the variables you exactly matched on, then "regression adjustment after exact one-to-one exact matching gives the identical answer as a simple, unadjusted difference in means" (see https://r.iq.harvard.edu/docs/matchit/2.4-15/Examples2.html)