Yes, 350.org can receive emails.
Yes, 350.org seems to be a legitimate domain.
A record
None
MX records
ALT1.ASPMX.L.GOOGLE.COM
64.233.186.26
ALT2.ASPMX.L.GOOGLE.COM
209.85.203.26
ASPMX.L.GOOGLE.COM
173.194.204.26
ASPMX2.GOOGLEMAIL.COM
64.233.186.26
ASPMX3.GOOGLEMAIL.COM
209.85.203.26
ASPMX4.GOOGLEMAIL.COM
74.125.140.26
ASPMX5.GOOGLEMAIL.COM
108.177.127.26