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Here are the answers:

1. There are 17 such state capital cities using population from the 2010 census). The biggest surprise to me was Columbus OH being quite a bit larger than Cleveland or Cincinatti or Dayton. Columbia SC is a little larger than Charleston SC and appears to be the closest qualifier although it’s similar to Charleston vs Huntingdon WV.

The full list (from east to west roughly)...

Boston MA, Providence RI, Charleston WV,
Columbus, OH, Atlanta GA, Columbia SC
Jackson MS, Indianapolis, IN, Des Moines, IA
Little Rock AR, Oklahoma City, OK, Denver CO, Cheyenne WY
Salt Lake City, UT, Phoenix AZ, Boise ID, Honolulu HI.

(bonus: according to the 2010 census, the closest second place capital city in percentage terms is Montgomery AL

(other contenders you may have considered)

AL: Montgomery 205764 .. Birmingham 212237 (97%)

TN: Nashville 601222 ... Memphis 646889 (93%)

CT: Hartford 124775 ... Bridgeport 144229 (86%)

MN: St Paul 285068 ... Minneapolis 382578 (73%)

LA: Baton Rouge 229493 ... New Orleans 343829 (66%)

NC: Raleigh about 60% of Charlotte, Bismarck ND not quite 60% of Fargo. Richmond is less than half the population of Virginia Beach and is also smaller than Norfolk VA.

(2) 41 of 50 have domes prominent enough to get a pass but maybe three others have some small dome feature so the right answer could be 41 to 44, some are quite a bit different in appearance than the U.S. capitol, for example Oregon has more of an inverted bell tower (maybe the craft beer played a role here), and Wyoming’s dome is not that prominent in comparison to the building itself, but of the nine or so that are nothing similar at all, three are what you might call skyscrapers (LA, NE, ND) and another three could be described more as temple style than capitol style.

AK a fairly ordinary small office building in appearance, think there’s a totem pole to add a bit of flair.
NM has the native longhouse look, quite distinct.
VA mega-courthouse like, looks like it belongs more in Rome perhaps. TN fairly similar.
AZ has a bit of a dome maybe, wouldn’t count it as similar to the DC Capitol at all, and just an edifice unlike the rest.
FL has both a dome and a skyscraper to cover all bases. The skyscraper is close enough to the domed portion to give a completely different appearance but without it, would count it as domed.
HI to my eye looks like an apartment building carpark, suppose it is vaguely Greek, that’s where O got his idea of a temple from perhaps.
NV has a vague dome feature but it looks more like an astronomy building on a small university campus to me.
NY has no dome, just the 18th century Amsterdam sort of style giving a hint of its age perhaps.
NC has a very hidden dome on a courthouse type building, not counted either.
OH has a different looking dome than all the rest, circular but flat topped.

Pictures here:

http://en.wikipedia.org/wiki/List_of_state_and_territorial_capitols_in_the_United_States

These look most similar to the federal Capitol: AR, CA, ID, ME, MN, OK, PA, UT (maybe KY but the picture is rather poor quality, a silhouette so hard to say).

(3) was a trick question, Springfield IL (pop 116,250) is the state capital of I’ll Annoy.

AL, AK, AZ, CA, CT, DE, HI, IN, KS, MD, MS, MT, NC, ND, NM, OK, TX, WA, WY _ none listed (a few others that I failed to note have no Springfield either)

There is a Springfield Center NY which technically does not count either. West Springfield in Fairfax Cty VA is also not counted here.

These states have a Springfield listed in the 2018 Rand McNally road atlas, approx locations, road distance, and in distance from any state capital order:

MA _ Springfield (pop 153060) while over 100 miles west of Boston MA is about 25 miles north of Hartford CT.
TN _ Springfield (pop 16440) is 25 miles north of Nashville.
AR _ Springfield (pop 185) 35 miles north of Little Rock.
LA _ Springfield (pop 487) 35 miles east of Baton Rouge.
NE _ Springfield (pop 1529) 38 miles east of Lincoln (south of Omaha and not far from Wahoo !!!) (nuclear power plant?)
SC _ Springfield (pop 570) 40 miles southwest of Columbia.
NH _ Springfield (pop 80) 45 miles northwest of Concord.
KY _ Springfield (pop 2515) about 50 miles southwest of Frankfort and (hmm) about 50 miles south of Shelbyville KY. Wonder if they have a nuclear power station?
MI _ Springfield (pop 5860) is a suburb of Battle Creek and about 52 miles southwest of Lansing.
OH _ Springfield (pop 60608) is about 55 miles west of Columbus.
OR _ Springfield (pop 59403) is the northern suburb of Eugene and about 60 miles south of Salem.
VT _ Springfield (pop 3989) is about 85 miles south of Montpelier in southeast VT.
MN _ Springfield (pop 2152) is about 90 miles w.s.w. of St Paul.
NJ _ Springfield (pop 14425) is near Newark and about 95 miles northeast of Trenton.
FL _ Springfield (pop 8903) suburb of Panama City 120 miles from TLH.
PA _ Springfield (Township) (pop 24211) is a suburb of PHL and about 130 miles from Harrisburg. Another Springfield is a hamlet near the NY border a similar distance north.
WV _ Springfield (pop 477) is in the northeast panhandle and is about 140 miles from both Charleston and Harrisburg.
MO _ Springfield (pop 159498) is about 150 miles southwest of Jefferson City.
SD _ Springfield (pop 1989) is on the Missouri River about 170 miles southeast of Pierre, probably about the same distance to Lincoln NE.
ME _ Springfield (pop 120) is about 180 miles northeast of Augusta.
GA _ Springfield (pop 2852) about 200 miles southeast of ATL near SC border.
ID _ Springfield (pop 100) about 240 miles east of Boise.
CO _ Springfield (pop 1451) 300 miles southeast of Denver and a bit closer to Santa Fe NM.
WI _ several small Springfields all over the state

I wonder if there’s a Homer J Simpson in any of them? The total population of all of them is over half a million so that you could say 0.1% of all Americans live in Springfield somewhere or other.

disclaimer — this info is moderately accurate in general, I sort of ran out of time and frankly near the end of the Springfields, interest. D’oh.