Sparatic starting and not starting.

On a very cold morning truck wouldn't start. after
trying for several minutes, 10 or 15, it started to
sputter. I tried it a couple more times and
eventually it turned over. it cranked the entire time i
was trying so not thinking its starter.
after work same day it wouldn't start. tried the
same thing I did earlier that morning, ultimately
had to jump start it, 15 minutes of charging off of
another vehicle, and it started right up. bought new
battery that same night, a lower cost affordable
Budget Battery, battery was replaced and installed
by auto parts store and it started right up. with new
Budget Battery next morning tried to start it,
wouldn't turn over cranked entire timr but never
sputtered. eventually jump-started it off of a
separate car without charging at all. it started right
up after work that day got into truck turned it over
and it started right up ?.
Went to get a much better Battery. something with
better cold cranking amps. explained in detail to
auto parts store what had been happening.
they tested the current budget battery, it checked
out. before the test however, upon my arrival to the
auto parts store I turned my truck off.  after battery
test clerk asked to start my truck back up so that
we could check the alternator volts. the truck
would not start with  budget battery. IT JUST
STARTED MINUTES AGO. this auto parts store is 9
miles away, I beg your pardon 9 minutes away
from my office so not very long at all was the
vehicle running. we try to jump box to start it. that
did not work we tried a high quality jump box. that
did not work we pulled the budget battery out put
the higher-quality battery in. brand new and
attempted to start the vehicle. it did not start
thinking  alternator.  we eventually jump-started it
off of his personal vehicle. after charging for 5
minutes or so it started up. we check the voltage
on the alternator. the alternator has 14 volts so
good on the voltage. we check the diodes those all
past. I left there with an upgraded battery and went
to get a second opinion at a different auto parts
store, also to check my core that I had replaced at
that particular store.  upon my arrival at that store,
I left truck running. I explained what was going on
to the clerk. the clerk test the battery, the upgraded
version, it checked out. check the diodes they all
passed. check the alternator and it was at 14 volts
or so, so all good thereď obviously very confused
as to what is going on at this this point!
Went to back to of the warehouse to test my
original car battery it checked out as a good
battery. at this moment I'm thinking that I have
another issue not a battery, so I refunded the
upgraded battery and ask to put my original core
battery back in, obviously very dead the battery
would not start my vehicle after charging for 15
minutes it's started. I put some seafoam brand
engine fuel system cleaner that evening, hoping
that perhaps my fuel line was the issue fuel pump
something of that nature the next morning I started
my truck and it started right away. Hallelujah!, I
thought....
after a half day of work I went to get in my truck
and start it.  it would not start.. very confused.do I
have two separate problems here? has this kind of
thing happened before? does anyone know where I
should start? people have been suggesting I have a
fuel pump problem a fuel line problem an air intake
problem a fuel filter problem. but that wouldn't
explain why a jump-start is helping. can anyone tell
me what's going on? Sorry for the long post but it
was necessary.