Yes, yes, no, maybe, yes & No
Well, not a broken toe but 11 days after sustaining a chip fracture on my foot, I did do the classic loop (Needed?? Lafayette as final 5K peak in Northeast at the time & did Lincoln for 2nd time that year) Doable, depend on how much your toe hurts, how you deal with pain, your boots & how you will immobilize the toe.
For the chip fracture I wore the stiffest boots I had (which were not that stiff - I now have EMS replacement Traverses which when I sold them for EMS we told people that their feet change to fit the boot not the boot breaking in)
Also taped it well (I'm familar with quality ankle & knee wrapping from years of playing Soccer, VB & BB) & this was my first hike using poles. The loop still took 10 hours to do & to this day, the foot still bothers me almost 9 years later. (okay so I took two days off after my hike, then did Osceola's & the day after that some CT AT mileage but I never said I was that smart either.)
Yes, you can hike with broken bones, why would you?
Next your brother is 15, is he active? I know 15 year olds who would eat the entire ridge up in no time at all & some that would be huffing & puffing long before the first view on the OBP? Which is your brother?????????
If you are looking at being macho, IMO, Welch Dickey is kind of lame, I'd look at doing Osceola's instead. Easier, lots of rocks early on from Tripoli Road which will test your toe pretty well for flat terrain (make sure your boots feet well because if they don't & your toe is sliding into the front of your boot, you will hate yourself big time) & the chimney between the two Osceola's while short will provide enough wow & this is hard factor (especially if you don't take the by-pass) to make any 15 year & his older brother feel good. This is better than trying the Ridge in your condition with 15 year old with no experience.
Info on the ridge:
1.7 miles from Little Haystack to either Liberty Springs Trail or Liberty Summit from there another mile or so to Flume. No way a cripple & neophyte have any business descending Flume Slide Trail, probably should not go up it either.
On Franconia Ridge Trail, steepest terrain is just south of Little Haystack & in your condition & your brother's ability (unknown or none you'll find out) your best bet is to go up Liberty Springs, hit Liberty & then ascend Little Haystack over the others & down OBP. Then have the 15 year old jog back to car (you'll be a couple of miles away from your car) & drive up to save your broken toe two more miles of walking downhill.
Is telling parents your underage little brother was in a car accident while driving any worse than perhaps requiring a rescue on Lincoln or Lafayette, I don't know, which one is more likely depends on the weather I guess. Either way, I'd dread the beating once I got home (even at 42 now & my Dad at 66) worse than the foot injury.
Either wait until you are better or pick an easier hike. Leave dumb hikes to dummies like me.