I think this is a pretty simple concept. I have one sound card and one pair of bluetooth headphones. I want to play my audio through both my sound card and my bluetooth headphones.

I believe Windows has checkboxes allows you to "check" outputs to enable/disable them, but Ubuntu seemingly has the equivalent of radio selectors (you can only select one at a time).

Bonus Question:
On a similar note, I have 5 analog output channels on my sound card (in addition to my digital & HDMI audio) -- I would like to be able to determine what comes out of each of those ports (e.g. "front speakers" on all 5 or "front", "center", "back", etc).

Thanks in advance!

With paprefs you have access to a virtual output device that enables simultaneous output to all attached sound cards/devices:

The additionally created audio output device for simultaneous output may be selected in the "Output" tab from pulseaudio sound preferences menu:

In this example it is shown for a HDMI-device but as soon as your Bluetooth device is recognized it will also be available for simultaneous output.

The changes may need a restart of pulseaudio to take effect either by logging out and back in to your session or by running pulseaudio -k in a terminal.

Ubuntu

I've just confirmed this solution (from 4 years ago) still works on Ubuntu 14.04 LTS.

Install

Open terminal and type sudo apt-get install paprefs go to the tab exactly as per the picture above and select the option.

Initial Run

Then; remaining in terminal, type pulseaudio -k to kill and restart pulseaudio.

Then go to your sound settings and you will see the option to output to multiple sound devices.

Props to whoever wrote paprefs it's a brilliant little piece of software I would actually like to see included in Ubuntu without requiring additional installation.

Apple Macs OS X

A similar solution is available via an included piece of software and whats so great about pulseaudio is that the sound seems to be perfectly in-sync from both outputs so it must be adjusting for the lag as well which is why its so impressive; otherwise we would be hearing a slightly delayed version from one output and another.

To complement the excellent answer from Takkat, I found the default name given to the new device was excessively long and distorted the Sound Settings dialog. In order to shorten that name, I had to additionally execute the following command:

gconftool --set --type string /system/pulseaudio/modules/combine/args0 sink_properties=device.description=Combined

I could not get Takkat's proposal work out of the box in Debian 8.7, although I restarted the system.
I assume you have completed Takkat's proposal in installing paprefs.
Extension on Takkat's answer which works based on Arch Linux wiki where keep analog input and Pulse calls that "duplex"

# /etc/pulse/default.pa

# http://unix.stackexchange.com/a/180374/16920

load-module module-alsa-sink device=hdmi:0

load-module module-combine-sink sink_name=combined

set-default-sink combined

Then restart pulseaudio:

pulseaudio -k

In Kubuntu 18.04, Plasma 5.12 paprefs is not needed, as a similar setting is already there:

A new output option should become available after reboot, called “Simultaneous output”.

Or in pavucontrol:

In 18.10 with Plasma 5.13.5 that “Simultaneous output” option has been removed, so aprefs is needed.