I go on many dates, and I always initiate. Women in my experience do not ask for a date - even if they are sending the initial message on Tinder, for example.
I think it's unfair to put the burden of paying on the person who asks, as it is almost always the man.
What kind of non-drink/dinner settings would you suggest? In my experience they don't go over as well as more traditional first dates. (Parks, etc.) Even the museum costs money nowadays. Hardly anything that is appropriate for a date is going to be free.
I asked a few guys out back in the day, but (as I said) I recognize that guys by far do the most asking. As for non-drink/dinner settings, (been asked out/asked out ):
Local festival/fair
Shakespeare in the park
BBQ
Beach concert (free)
Paint and Drink (was free on a Groupon when I went, but there are plenty of other random 'free' things you can finagle somehow)
Hiking
Picnic
Scavenger Hunt (was a local event)
Biking
Poetry reading (I did not personally enjoy this, but it was 100% free and some people are into that)
Museum (our entry was free, and most museums have special free days or events, so...)
On the non-dinner low cost/low pressure side of things:
Pub Quiz Night
Rock Climbing
Drink and Draw (my date idea)
Wine tasting (if your non-drink query just meant 'not at a bar' kind of place)
Laser tag (yeah, I know, I'm lame. I just really like laser tag...)
Arcade (I prefer barcades)
The range (ammo did go up quite a bit, though, so kinda middle-range?)
I mean, unless you're pretty far out in the boondocks, it seems like there's always something on, or at the least something silly you can do. And even if museums cost money, that's far cheaper than a dinner and a lot less pressure, right? Dinner or a bar doesn't have to be the default unless that's what you're most comfortable with. I certainly had far more fun on the random adventures that some guys suggested.