Hi Assaf,
I hope I can help.
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Weights are briefly summarized on the FAQ page. In general, you should use WT06 together with the replicate weights found in RWT06 for empirically derived standard errors. There is sample code for replicate weights on the CPS replicate weights page. You should use WT20 together with RWT20 when analyzing the 2020 sample. All of the weights are probability weights. In stata, you would run your regression with [pw=wt06].
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Finding total time spent with children can be tricky. You’ll need to use AGEW to examine the ages of people the respondent engaged in activities with. Summing the number of occurrences of AGEW < 18 and tabulating them by HH_CHILD, you should find that among ATUS respondents in 2019-2021 with household children only about 5% did not engage in any activities with children under 18. Taking it a step further, respondents engaged in primary and secondary childcare sometimes omit that they were with a child at the time. When including households with ACTIVITY codes 30100-30399 (primary childcare for household children) and positive values for SCC_HH_LN (secondary childcare for household children), you should find that less than 5% of respondents in households with children report not being engaged in either primary or secondary childcare or any other activities with those under 18. This seems reasonable considering the respondent is a random member of the household.