How common is drawing Л the way similar to Greek letter Λ in modern Russian? Does it vary depending on region?
In short: angle-shaped Л is common to texts written by the hand (handwriting and block letters). In fact non-angle-shaped Л is not used at all in such cases.
For printed fonts non-angle-shaped Л is common (roughly 90% vs 10%), though it may be used for artistic effect or to add classic-style flavor to the text.
Actually angle-shaped Л is always used when you write block letters by the hand. It is because it is much simple to draw. (In fact, maybe this is the reason why "hand written" MS Comic Sans font looks unnatural to me, because they use Л instead of anlge-shaped letter).
Here is how the children are taught to write this letter (taken from question about italic т):
The Russian Wikipedia states that current non-angular shape of the letter was first introduced back in 1840 and later become very popular in Russian typographics. Though in many artistic fonts the angular-shaped letter is still used. For expample here you can see the titles of Petersburg subway stations. Most of them use the angle-shaped letter.
Here is a post about shapes of Л by Юрий Гордон, the author of the book Книга про буквы от Аа до Яя. It is interesting to note that in most fonts when the angle-shaped Л is used, then Д also has such form. And when Л is more like П then Д also has similar form.
As for printed media - some academic books use fonts with angle-shaped letter, for example "Литературные памятники" series that was established back in 1951. Here is an example of covers of books printed in 2012, 1983 and 2011 (last one is from some other series):
As seen in the street:
Fonts with such letter were popular back in 50-s (because of Constructionism art movement that preferred simple geometric forms), so you can frequently see it on objects created in those times, or ones that have retro-design:
Well I've never seen it in printed media
and I've never met a font using this shape for «л». I'm afraid it will be a bit difficult to recognise, but, well, as part of a logo it might be fine (I don't remember any logos with Λ though).
Also note that it is more common to use it instead of «A» in logos.
Update. Here are some fonts: