Am I still a LPR while abroad with the re-entry permit?

moray

Registered Users (C)
Basically, this question has two aspects: travels and taxes.

Travels
My understanding is that while one has a valid re-entry permit one can enter and leave the States as much as he/she needs without getting new re-entry permits every time, correct? I have read a lot about these permits and this question somehow is never answered straight.
Also, one can still enjoy some LPR benefits like visa-free entry to Mexico, right?

Taxes
All the time one is abroad and has a valid re-entry permit (technically - while his/her Permanent Resident Card is valid AND one's residency is not considered as abandoned) he/she is considered a resident for U.S. federal tax purposes, as far as I understand. Meaning a need to file 1040 on the worldwide income and complying with FBAR requirements.

Thank you.
 
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