I am a US greencard holder, got greencard through lottery. I lived in the US for three years, and I am currently living in Europe where I came to join my husband. I know that if I intend to apply for a citizenship I need to come and stay in the US for a month every six months to continue my residency. Is there any way around it?
That's wrong.
You should be living and maintaining residence in the U.S. , not for naturalization, but also in order to keep your green card valid. Coming for one month every six months is not enough. If you stay more that one year outside of the U.S. you may get you green card taken from you at airport upon your arrival, and yourself being deported (sent back) to where you came from.
CIS has a list of reasons why a green card holder may be absent from U.S. for longer than one year, e.g. studying abroad, being sick, etc. You would need to obtain a reentry permit if you stay outside of U.S. for a long time.
As for citizenship, the basic rule of thumb is that you need to live for five years in the U.S. before applying for naturalization.
Also, do i need to collect any evidence of me being in the US for a certain period of time, or is this information registered somewhere anyway every time I cross the border?
The information about your coming and leaving does get registered with CIS. When you depart, you give your green card to airline company rep, who scans it. The scanned info goes into passenger list called manifest, which gets sendt to various law enforcement agencies including CIS. When you arrive and come to the immigration officer for inspection, he scans your green card and verifies its validity online, while also making a record of your arrival in CIS database. If you get called into CBP room for whatever reason, all information about your trips can pulled out from CIS computers. I know that for once I was taken for questioning at JFK when I flew in there, and immigration interrogator asked me about every one of my recent trips, citing when and where I was going to.
You however should collect your itinerary information on your own..
Most people keep the airfare tickets and boarding passes. If you use e-tickets, don't forget to make a printed hard-copy of every e-ticket, because once e-ticket is used up, it's info is not available online anymore.
Same for bus or train.
If you travel by car, you may consider keeping cash receipts issued by some state authority. Also, highway toll statements are useful.