Yours will certainly have a two tooth sector. As to the specific length, it may be best to buy your steering column first and then fit your parts around that according to what you measure. If you have the steering column you should be able to measure using a long piece of wire.
As you allude, there are markings on the side of the steering box which will tell you the specific Ford/gemmer unit. However, this is not as secure a determination since the upper part of the box/column COULD have been swapped out by a restorer looking to get a better looking worm. The fit for either brand-mark is identical into the top of the two tooth box and parts can be interchanged.
The headlights and light switch for a 1930 Model A should be of the "twolight" system and will afford a 4 position switch. Handle pointing to the lower half the wheel, positioning IIRC is park(left)-off (straight down) - low(right 1st click) - high (right 2nd click). Sometimes a wiring glitch can reverse the sequencing between low and high beams. The wiring glitch CAN (probably mostly does) occur in the headlights themselves. Assembly of the headlamp internals is not random though - original parts are keyed to go together in only one way. Repop parts not so much.
The horn rod is hollow and a single wire serves to "ground" the connection FROM the horn and complete the circuit. Kits are sold to replace the wiring. Also to replace the black plastic button which tends to get broken on the 30-31s. A short circuit in the wiring downstream of the horn will cause it to continually blow - a defect often seen with reproduction light switch housings or older cloth covered wiring.
You probably can remove the horn/switch rod inside your 30 sedan without further effort. I cannot on my 29 CC pickup. Still, it may be easiest to unhitch the two spark/throttle linkages, unbolt the column upper connection (2 bolts) and unbolt the lower connection to the car frame (2 bolts) - BUT NOT REMOVE THE COLUMN FROM THE CAR - Leave it attached to the steering linkage at the drag link. You can do your horn rod replacement with the additional headroom gained by dropping the column slightly which removing the bolting allows you to do easily.
Your former headlights sound very much like a typical 1940s "keep her running" mode. I think of the vast the number of dash trim panels that have been drilled to replace the not so reliable original Ford headlight switch with an aftermarket push-pull kind of switch.
But the original Ford switch has a lot to offer in convenience and originality.
Try to avoid repop parts if you can. But I hear they are getting better. Hoping anyway.
Good luck,
Joe K