I just did the OKC-KC drive yesterday; it's a regular trip of mine thanks to family in Kansas.
As to the primary question of your post, re free I-35 vs. Kansas Turnpike: It really depends on where in KC you're wanting to go as to which route is "better" in the conventional sense. If you're most interested in destinations on the west side of Kansas City, Kansas (i.e. the Legends/Speedway area), the Turnpike route is faster. If you're more interested in Johnson County or the Grandview/Lee's Summit area, free I-35 will be better. For downtown KCMo, it's probably kind of a crapshoot, but I would think I-70 would be faster since you get to bypass Johnson County. Traffic is pretty light on I-335, although that kind of gets canceled out by the heavier traffic on I-70 between Topeka and KC.
That said, whenever we go, we nearly always use free I-35, despite the fact that our destination is in west KC. For one thing, the toll from the southern terminal to Emporia alone will set you back $9.50, and it just gets higher from there. Also, the Flint Hills are interesting the first time you see them, but by the time you get to Emporia they tend to get kind of monotonous, so cutting east on free I-35 is appealing for the change of scenery it offers.
Also, if you stay on the Turnpike, because of the ticket system, you are kind of at the mercy of the KTA service plazas as far as availability of food and gas goes. On free I-35, Emporia and Ottawa are good stops. Our usual routine is to eat breakfast with a full tank and leave Norman around 10 AM or so; this puts us around the state line by the time we get hungry, and in Ottawa when we need gas. Stop for lunch at the Belle Plaine plaza if you prefer McDonalds (and free maps!) or Matfield Greene if you prefer Hardee's. Avoid Towanda at all costs; as the middle plaza, it is the busiest, and is often used by buses. If you continue on I-335, you have a few more options, but I've never stopped at any of them so I don't know what they're like.
There is a lot of resurfacing work on all of I-35 in Kansas at the moment, but it didn't result in any real slowdowns.
If you're mostly doing this trip for roadgeek interest, why not take one route there and the other back? Then you can see both of them for your own satisfaction.
Possible options for the KC-Omaha leg... You could depart Kansas City on US-73; it begins in west KCK and ends at US-75 just into Nebraska, so it's a cheap clinch of a two-digit US route. If you prefer the I-29 option, it may be worth it to detour onto I-229 either on the way there or back, as it includes an endangered double-deck section that parallels the Missouri River.
