you are allowed to travel, but can face Scrutiny from immigration for long lengths out of the country. realistically you’re not to be out of the country for more than 6 months. so you should be good. if they question you why you were out of the country for a month, give them your reason and as long as its a good reason they wont question you further

yes you can

i have a conditional GC, I travelled without my hubby and stayed 2 weeks outside the country. when I came back I was asked where I went, why did I go, how long did I stayed, what type of job I do in America and what did I took back. All thes questions are from immigration officer customs officer never ask me anything.

I traveled with Advanced Parole and I was very fine not to talk of GC. you’ll be fine. I spent one month.