1010tires.com has a tire size calculator, if you compared the two tires you'd get these results:
Stock Vehicle Tire Size:
205/65R15
Section Width 8.07 in
Rim Diameter 15.00 in
Rim Width Range 5.50 in - 7.50 in
Overall Diameter 25.49 in
Sidewall Height 5.25 in
Radius 12.75 in
Circumference 80.09 in
Revs per Mile 815.94 /ml
Actual Speed 60.00 mph
Speed Variance [Reference]
Diameter Variance [Reference]
205/55R16
Section Width 8.07 in
Rim Diameter 16.00 in
Rim Width Range 5.50 in - 7.50 in
Overall Diameter 24.88 in
Sidewall Height 4.44 in
Radius 12.44 in
Circumference 78.16 in
Revs per Mile 836.08 /ml
Speedometer Reading 63.67 mph (3.67 mph higher)
Speed Variance 2.47% too fast.
Diameter Variance -2.41 %
Notice the overall diameter of the new tires is 24.88 in, 0.61 in smaller than the original tires, so for one mile the new tires will spin more times, throwing off the speedometer (it will read 63.67 mph when you're really going 60 mph).
But, since the tire's smaller it should not interfere with anything on the car, and the bigger rim won't hit anything on the inside either. If the diameter were larger, or the rim size smaller, you could have interference problems.
The site recommends staying within 3% of the original size, much larger tires could cause braking problems (a "longer lever" effect).
So, the new tires should fit ok.
I'd notice that the Sidewall Height of the old tires is 5.25 in, but the new ones are only 4.44 in, 0,81 in smaller, so a rougher ride is possible too.
About the rear wheel being "stuck", if there's really no lug nuts still holding it on, the rim's probably just rusted & stuck onto the hub, I'd try a hammer on the sidewall of the tire (inner or outer) to knock it loose, and the tire's being replaced anyway. A pry bar would be a last resort, it could be very easy to bend something important.
If you're re-using your nephew's rims too you'll also have to check the rim's bolt pattern and offset, I don't know the year of the Civic, but not every Honda from every year will have matching bolt patterns. Just check a web search, or it might be written on the rim.
You should probably phone some local scrapyards too, they often have used tires in good shape and you could get the right size. If you're in America I wouldn't expect a used tire & rim to cost $88 either, especially from a pick-your-part type place a tire & rim should cost a lot less. For example, Pick-n-Pull in Modesto, CA charges $17 or $19 for a 15" tire, plus $7.99 for a 5-lug steel rim or $11.99 for a 6-lug steel rim (or $20.99 if an aluminum rim). So you could pay $25/tire for the right size instead, if you don't mind walking around & taking them off yourself.