This question is unclear. First, "duplicated" in what sense? If data for the same unit accidentally is repeated in the file, you should de-duplicate, but it might as well be two units with the same values. Two or more six year old girls with the same response value is not impossible ... If you do not know which case, it is up to you to investigate and find out.

Assuming the last case, this will not make any problems. The $X$ matrix itself is not inverted so that is not an issue. You can group the identical rows if you want, but it is not necessary.

R still calculates this regression - is it just throwing out duplicated cases?

R (and I hope other software) will never throw out duplicated rows.

And another question regarding the grouping process itself. If I use grouped data, I just have to throw out double rows and record the frequency and use that as weights?

With one row per unit, you have Bernoulli data, the response is 0 or 1. With grouped rows, it is Binomial data and the response is now the total number of successes, you must also have the total number of units grouped together as another variable. Let this variables be x, n . One way of specifying the response with the R formula language is now

cbind(x, n-x) ~  ...