The purpose of I and Q channels in RF communication
I and Q modulation does not double the bandwidth but rather allows you to send twice as much data on a channel (fixed bandwidth) as you can achieve with pulse-amplitude modulation
(PAM) on a single carrier signal (which gives a double-sideband (DSB) signal).  A quadrature amplitude modulation (QAM) signal is just two different PAM (baseband)
signals modulated onto phase-orthogonal carrier signals (the cos and sin as you noted).
Both signals are DSB signals.
The two carrier signals must be of the same frequency and must differ in phase by 90
degrees. The sum of the two DSB signals is the QAM signal, and it occupies the same
bandwidth as the two separate PAM signals. Because the required phase orthogonality is difficult to achieve (and maintain!)
with two separate oscillators in the same location, let alone different locations,
QAM is not a multiple-access method: the sum of the signals from two different transmitters each creating its own PAM signal and modulating it onto a carrier does
not give you QAM unless the two carriers are at exactly the same frequency and
differ in phase by 90 degree.  Thus, the answer to the question


  is QAM two independent channels that mixed with carriers of different phase or are the channels related?


is that QAM is not two independent channels and the carrier phases in the two
signals must be carefully controlled and maintained at 90 degrees.

The information in a PAM/DSB signal is carried in both sidebands but can be
recovered from only one sideband if need be. So, the bandwidth required to
transmit a PAM signal can be reduced by a factor of 2 by filtering the PAM/DSB
signal to get a single-sideband (SSB) signal. (Of course, one could create
a PAM/SSB signal right from the start instead of filtering a PAM/DSB signal).
A separate PAM/SSB signal could
be transmitted in the other (unused) sideband and each can be
demodulated completely independently of the other. But the sum signal
is not a QAM/DSB signal and the demodulation
technique is different. Both receivers will use I and Q demodulation
and each must filter the incoming signal to eliminate the unwanted
sideband which means that filters with narrower bandwidths (and sharper cut-offs)
must be used.  Two PAM/SSB signals, transmitted separately in the
upper and lower sidebands give essentially the same spectral efficiency
(bits per Hertz) as QAM but at a higher price. The advantage gained is, 
of course, that PAM/SSB signals can be used in multiple-access situations.