How often can I travel out of the U.S. with my greencard?

Yes, so long as you have an intent to reside in the US (evidenced by physical presence, filing tax returns as a resident etc.) you should be fine to maintain your green card.

Yes, and you do have the option to select commuter status. If any one trip is greater than 180 days, that would delay the date you are eligible to become a citizen.

As a green card holder, you must reside in the U.S. for the majority of the year to maintain your status. Extended trips abroad (over 6 months) could raise concerns about abandoning your residency. Given your schedule, consider applying for a reentry permit, which allows you to be outside the U.S. for up to two years without losing your status.

If you can stay in the U.S. for six months every year, you should be safe to maintain your permanent green card.

Thank you for your inquiry. There is no limitation to the number of times that you can travel outside of the U.S. once you have been granted your conditional green card, nor is there a limitation once you have the condition removed. However, there is a limitation on the amount of time you can spend outside of the U.S. on a continuous basis – if a green card holder is out of the U.S. for more than 1 year they could be deemed to have abandoned their status. Even an absence of more than 6 months would be problematic as the CBP could subject you to a more detailed interview when you return to the US. Finally, absences from the U.S. can impact the timing in which you would be eligible for future naturalization.

There is no limit on. The number of times you can travel. You should be mindful regarding the amount f time you stay outside the U.S on a single trip.