What terminal emulators are available for heavy terminal users?

Besides GNU screen I would recommend tmux. It is a terminal multiplexer like GNU screen.

I personally use gnome-terminal as my terminal emulator. Using gnome-terminal, I run GNU Screen (byobu to be specific) in order to work with multiple windows. You can become really efficient with it. I use GNU Screen because I often have long-lived terminal sessions that I can check on or come back to later.

gnome-terminal running byobu (showing only one window open)

You seem more interested in a graphical solution for managing terminal windows. For this, I highly suggest the terminal emulator Terminator, which supports multiple windows without the need for a multiplexer like GNU Screen. You can split it horizontally and vertically as much as you like and there are even tabs. If you just feel like running lots of terminals, Terminator is really good.

Terminator (showing multiple Terminator windows open)

I like guake, which is a drop-down editor (named after the console in games like quake). You press a button and it slides down, press another button and it hides away. Press another button to make it full screen, etc. It supports transparency, tabs, color... you name it. (You could run GNU screen on the inside to get more than one visible at a time.)

It's on the top of this screenshot, transparently overlaid some stuff I was doing. (Click for full size.) I think it looks good.

One note: terminal multiplexer (tmux, screen, etc.) is not the same as terminal emulator (xterm, gnome-terminal, urxvt, etc.)

Aside from a recommendation (mine would definitely be tmux and I've switched from screen a while ago) you asked about customisations.

Color coding for everything

Make sure to choose a terminal that can support 256 colors. You can test if it does by downloading a simple Perl script from http://www.frexx.de/xterm-256-notes/data/256colors2.pl. chmod +x it and run it (after of course looking briefly at its code).

Another thing is (if you decide to use tmux) is to make sure it sets the TERM environment variable correctly. Make sure to put:

set -g default-terminal "screen-256color"

in your ~/.tmux.conf file.

Easy to modify the aesthetics of the terminal (is it vain to want my terminal to look nice?) such as transparency, borders, etc.

I'm using urxvt after switching from aterm (I've used gnome-terminal, xterm and many others before). Both have support for transparency and are fast (I'm talking about being able to display a long stream of lines quickly). They're configurable with ~/.Xresources or ~/.Xdefaults.

Borders? I think it's up to your window manager to draw those, which in turn depends on the theme you use. Good luck.

I highly recommend Terminator.

Colour coding depends on the commands you use though. grep and ls both accept --color

I use gnome-terminal + byobu (a wrapper around screen)

I can heartily recommend tmux. Having come from the excellent Terminator, it took a week or two to get used to the keyboard shortcuts but I'm now fluent, and finding it much faster and much more productive than a GUI-based emulator.

Like screen it runs as its own self-contained instance and it's great to be able to close the terminal window on one machine and open it via ssh on another. When I do, I find all my windows and panes where I left them. Handy.

For colour-coordination I'd recommend tmux-powerline. It gives you a status line that is very pretty (and useful too :) ) and once you have it set up, you can use vim powerline which uses the same resources (see screenshot for example).

I agree with Martin's answer regarding setting your terminal colours to screen-256color. This setting also seems to sort out occasional display issues between tmux and mutt, for example.

I suggest Tilda with Screen; also zsh as a shell could be a good thing!

For questions about related/similar/analog software you can check nice wiki alternativeto.net