Breathing removes CO2 generally. But when you are breathing normally, i.e. letting the autonomic nervous system control the breathing rate at tidal volume (the amount of air exchanged in a normal breath), the rate of purging CO2 is equal to the rate it is produced by cellular respiration so you have a normal physiological amount of CO2 in circulation.
If you intentionally breath deeply, greater than tidal volume, it means at the same breathing rate, you are exchanging more air in total and will be purging more CO2 than you produce which will lead to hypocapnia over time.
Hypocapnia doesn't inherently need to result from deep breathing if you offset the greater-than-tidal volume with less-than-normal breath rate — if you are breathing very slow or 'skip' breathing with pauses in between, but that balance can be pretty hard to reach
Depends on the balance of deep breath and breath hold. At the end of the day if the overall air exchanged is greater than necessary to maintain balanced CO2 levels, for example if you are deep breathing and pausing in between but not long enough to offset the amount of air exchanged, you could still be driving a net decrease in CO2 levels. However if you are breath holding until the urge to breath arises, which is driven by rising CO2, then yeah you are probably offsetting the effect of the deep breathing.