Hehe yea it seems that way but I'm not too worried

a) games like this are used for charity or corporate events. They're all casuals who have no idea what a sucker bet even is.
b) that config isn't final I'd probably tweak it a bit to make all the house edges roughly similar
c) the end result won't make use of colors, this was just for visual representation, I'll probably used numbered horses

Awesome, playing around in python made this super easy

I think I found my new game board:

current run		True odds	Odds to offer
red	32.40%	2.1	2
green	23.36%	3.3	3
yellow	17.13%	4.8	4
blue	13.82%	6.2	5
orange	8.00%	11.5	10
purple	3.42%	28.2	25
grey	1.87%	52.5	50

If I wanted to make this game more complicated and include Place and Show bets, could my simulation wizards to help me determine the odds of each horse "placing" (1st or 2nd) and "showing" (placing 1st, 2nd or 3rd) working from my above wheel configuration?

For this purpose we can still call the winning horse the one to cross the finish line first, but we can continue to resolve movements from the winning spin (if any).

The other constraints are that the next horse closest to the finish line (or second to cross the line) would come 2nd, the next would come 3rd.

Standard rules for resolving ties. If there are ties for second, there is no 3rd place and all 2nd place finishers "place". If there is only one 2nd place and multiple horses finish 3rd, they are all "place" finishers.

Is this doable?

this is more complicated than just finding the winner. there are more options that can happen with placing. there is only 1 winner, that never changes. but for other colors, what if they have something like [6,6,6,5,..]? or [6,5,5,5,..]? would the order of runners getting to these number of steps matter or not?

No, but I'll make it easier. I probably won't want to include show betting (third place). So I really only care about 2nd place now.

There can be ties. If 2 or more horses end up at 6 by the time there's a winner, that's fine.

If it makes the math easier, I can make the rules the same as first place; as soon as a 2nd horse finishes the race, it comes second and we stop?

yes, this would be simpler and it would be consistent with the rule for the winner. also, you would have to pay out only one runner every time

what you need to keep in mind is that it can happen that horse that wins comes from 4 to 7 steps and the second horse comes from 6 to 8 if you're counting it like this.

yea, wouldn't be a problem. "crossing" the finish line is all we care about, not how far they crossed it