Not really! I use an XCompose file derived from the Plan 9 keyboard setup. So to write ∏ i press my modifer (left shift) button once, and then 'p' followed by 'r'. So it is fewer key presses than \prod (three key presses vs five).

I have similar shortcuts for all maths symbols I use regularly. Here is an excerp from the configuration

2200 fa ∀ for all

2202 pd ∂ partial differential

2203 te ∃ there exists

2205 es ∅ empty set

2206 De ∆ increment

2207 gr ∇ nabla

2208 mo ∈ element of

(…)

220E qed ∎ tombstone

The last one is most sattisfying to type !

Not necessarily -- with the right editor, e.g. Emacs, you can have it automatically convert "\alpha" to "α" (built-in TeX input mode), or if you're using AUCTeX, you can type "`a", which would be easier than "\alpha" or "α"!

Edit: Oops, saw the sibling post. That also works, although sadly I can't take always advantage of that since I'm not always on Linux.

There's a very simple emacs extension called "pretty lambda", which displays a word "lambda" as a lambda character (but presents as ascii still). I wonder if something like this is possible to do here (of course, the syntax with arguments makes it harder).