2007 Chevy Tahoe LTZ

My '07 tahoe has recently been
giving me trouble on occasion to
start. It's not everyday, just
sporadically on occasion it will
go from a full charge(15v) and
starting fine when I leave, to not
starting when I try to go home. If
I jump it from another car with
cables it starts no problem. I
tried disconnecting the battery
cables while running and it
stayed running, which I was told
it was the battery if it stayed
running. The battery is a good
battery and looks brand new with
the cables clean and free from
debris. I just bought my truck
and little over a year ago and had
full maintenance done before I
drove off the car lot. It was a 1
owner car and very well taken
care of and I just don't
understand why it would go from
never having a problem to
suddenly not starting so quickly.
Like I said, I leave my house and
it starts fine and has a full charge
but when I get in to go home it
suddenly won't start. Sometimes
it won't do anything at all, no
clicking or trying to turn over, but
the inside lights come on, and
other times it will click a few
times but then nothing and
completely dead and for some
reason locks my doors but
nothing else. Also my key remote
has stopped working on some
buttons but works on others. It
started with the back hatch
button that suddenly quit
working completely, and now it
won't lock or unlock sometimes
and I usually have to tap it on
something hard a few times
before it starts working again,
but the hatch still doesn't work at
all. Any help would be greatly
appreciated. I've been reading
over all previous posts on
anything close to mine but would
really like clarification. Thanks