The Republicans in the Senate tend to come from really small states that represent much fewer people than the Democrats who tend to come from larger states.
You're right about the other points, but the Senate claim isn't really true. The 10 lowest population states are represented by 10 Republicans and 10 Democrats. The 10 highest population states are also represented by 10 Republicans and 10 Democrats.
No, still true. Here's an analysis using 2012 population data (easiest available) and the current U.S. Senate seat occupancy, assigning the independents to the Democratic tally since they caucus with the Democrats. For simplicity each person gets counted twice (once for each senator that represents them), and I only tallied the number of people represented by Democrats. The 3rd column is the state population if they have at least 1 Democratic senator. The 4th column is the state population if they have 2 Democratic senators. A 0 means the senator is a Republican.
State St Popn St Popn D1 St Popn D2
Alabama 4,822,023 0 0
Alaska 731,449 0 0
Arizona 6,553,255 0 0
Arkansas 2,949,131 0 0
California 38,041,430 38,041,430 38,041,430
Colorado 5,187,582 5,187,582 0
Connecticut 3,590,347 3,590,347 3,590,347
Delaware 917,092 917,092 917,092
Florida 19,317,568 19,317,568 0
Georgia 9,919,945 0 0
Hawaii 1,392,313 1,392,313 1,392,313
Idaho 1,595,728 0 0
Illinois 12,875,255 12,875,255 0
Indiana 6,537,334 6,537,334 0
Iowa 3,074,186 0 0
Kansas 2,885,905 0 0
Kentucky 4,380,415 0 0
Louisiana 4,601,893 0 0
Maine 1,329,192 1,329,192 0
Maryland 5,884,563 5,884,563 5,884,563
Massachusetts 6,646,144 6,646,144 6,646,144
Michigan 9,883,360 9,883,360 9,883,360
Minnesota 5,379,139 5,379,139 5,379,139
Mississippi 2,984,926 0 0
Missouri 6,021,988 6,021,988 0
Montana 1,005,141 1,005,141 0
Nebraska 1,855,525 0 0
Nevada 2,758,931 2,758,931 0
New Hampshire 1,320,718 1,320,718 0
New Jersey 8,864,590 8,864,590 8,864,590
New Mexico 2,085,538 2,085,538 2,085,538
New York 19,570,261 19,570,261 19,570,261
North Carolina 9,752,073 0 0
North Dakota 699,628 699,628 0
Ohio 11,544,225 11,544,225 0
Oklahoma 3,814,820 0 0
Oregon 3,899,353 3,899,353 3,899,353
Pennsylvania 12,763,536 12,763,536 0
Rhode Island 1,050,292 1,050,292 1,050,292
South Carolina 4,723,723 0 0
South Dakota 833,354 0 0
Tennessee 6,456,243 0 0
Texas 26,059,203 0 0
Utah 2,855,287 0 0
Vermont 626,011 626,011 626,011
Virginia 8,185,867 8,185,867 8,185,867
Washington 6,897,012 6,897,012 6,897,012
West Virginia 1,855,413 1,855,413 0
Wisconsin 5,726,398 5,726,398 0
Wyoming 576,412 0 0
Total Democratic: 334,769,533 ~53.5%
Total Republican: 291,793,901 ~46.5%
Democratic Senators: 46
Republican Senators 54
So Republicans are over-represented in the Senate as well (by about 7.5 seats).