It's very hard to say as it depends also what you expect fron commuting, which fleet and where exactly in EU you want to go.
For 787/777 (and minimum required by rules thus could be applicable to any fleet which is less busy such as 320)
You get a block of 4 days off every month. The other off days are scattered over your month. The off block is part of a legal rest, most likely after a trip that needs 2-4 local nights off so you likely will be tired because kf significant jetlag. The 4 days off block will not be given if you have annual leave or call fatigue a few times.
Rosters have 7-8 days off in Doha in total. Sometimes a few more are given but don't count on it.
If you have leave your normal days off are reduced, example if you take 8 days of leave expect 3 off days more that month essentially making 8 days of leave only 3 off days more than you would have anyways in a month.
Bidding for 6 days off or extra days with annual leave is possible but expect very few chance of getting it.
Regarding covid you need to do a rat test after comming back from your leave days. Be aware you need to be in doha always 12h before duty.
I believe it's not possible to commute in QR, 4 days is not enough and your arriving and departing flights need to fit very well to make these 4 days off at home possible.
Unfortunately bringing your family to Qatar isn't ideal either as you'll be only 8 days in Doha and be very tired since you'll easily fly 120 block hours every month.
Best of luck.