The simple answer is no (to the 21 speed vs. 7 speed question having similar minimum range). The better answer is yes if the single front chainring matches the # of teeth of the smallest chainring on the 21 speed.
Remember that with a 20" diameter tire, even a 1:1 ratio (20 gear inches) is a decent hill climbing gear. For example, on my 21 speed bike, I have 26" diameter tires and a 24/28 lowest gear so that is about 22.3 gear inches so if you had say a 34/34 setup (with 20" tires), your gear would be even lower than mine. To beat 1:1 on a 20" tire I would need 24/32 gearing on my bike which is possible but would cost me some money to get. I would say shoot for 20 gear inches and go from there.
Whoever recommended borrowing someone else's bike and testing that first seems like good advice. That will give you a baseline to tweak your gearing from.
What you may want to try is gear the bike so you can normally make the hill in the 2nd to lowest gear, thus giving you 1 more bailout gear for when you are extra tired. Condition yourself to be able to make it up in 2nd gear but you will have the assurance of having 1 lower gear. For example, instead of a 34/34 setup being your lowest gear, you can make it so you have 28/28 as your next to lowest gear, then you will have 28/34 as your bailout gear, which with 20" tires, would give you about 16.5 gear inches.
Finally, yes I think if you don't go ultra low on the lowest gear, then a wide range 6 speed should also suffice for normal "flat" riding and coasting downhills. For example, 28 teeth in the front and 11,14,18,23,28,34 in the rear would give you a gear inch range of 16.5 low to about 51 high. Just F.Y.I. (and for reference), my favorite and most used gear on my 21 speed is 34/20 which with my 26" tires gives me 44 gear inches so even if your smallest cog is 13 (not 11), you will be at 43 gear inches in top gear, good for a leisurely 9 to 10 MPH (14.5 to 16 kph) pace comfortably.