Way2slow

Should be easy enough to figure out.  If you have an ohm meter you can do it just by doing resistance checks.  If not you should be able to just use one battery, but see if this helps you any

Single or dual cable?  Single cable, I haven't seen many of those you can adjust so the steering wheel doesn't have free play at the helm.  Some used to give you and adjustment on the helm, but it's been so many years since I've even seen a single cable, I wouldn't have a clue what they do now.  Dual cable is not a problem, you just adjust the housing where they attach to the motor.  I would first make sure it's the cable itself and not something else.  
I've seen a lot of them the motor mounts and cable connection points have a ton of movement. 
One other thing, make sure it is cable and not hydraulic.  I had a guy ask me one time if I could adjust his cables because he had a lot of free movement.  When I looked at it, told him that was not possible, but I could bleed his hydraulic steering system for him and probably take care of it.
 

AGM require less maintenance.  They generally have a longer cycle count (service life) but will not have the Reserve Minutes/Amp Hour capacity of a similar size flooded cell battery with service caps on them.  With both AGM and Maintenance free batteries, they have to reduce plate size to make room for the glass mat material in AGM or extra electrolyte in maintenance free non AGM batteries.  The less lead in the battery, the less capacity the battery has, and the lighter the battery will be.  AGM's are a little more sensitive to the charging voltage and maintenance float voltage but unless you charger is extremely old, it's probably suitable for AGM.  
Now, with all the said, I usually don't buy AGM trolling motor batteries because of the reduced capacity.  Don't let the higher MCA/CCA miss lead you.  That means absolutely nothing when used as a deep cycle TM battery.  That's stickily just extra ump it has for cranking the big motor. 
You want to look at the Reserve Minutes (RM), the higher that number, the more run time you will get on the TM.  If it does not list the RM, then look for the higher Ah capacity.

It's basically just a Mercury Tilt/Trim pump.  
https://www.bing.com/videos/search?q=how+to+add+fluid+to+mercury+trim+pump&&view=detail&mid=D06C876CE0B5A1278F71D06C876CE0B5A1278F71&&FORM=VRDGAR&ru=%2Fvideos%2Fsearch%3Fq%3Dhow%20to%20add%20fluid%20to%20mercury%20trim%20pump%26%26FORM%3DVDVVXX

4A M 6E0-063065 - 6E0-076814 1985

When doing the math on battery capacity to figure run time.  There are two variables you have to take into consideration.  A 10Ah battery is rated at,' from the time it's fully charged until is reaches the point of being fully discharged.  That's usually in the lower 11 volt range.  Your electronics are not going to stay on until reaches that fully discharged level.  Many won't even stay on until is at 25 to 35 percent charged.  So, you have to figure in how long it's going to run until it cuts off, not how long is before the battery is fully discharged.
That's where LiPo batteries have a major advantage over lead acid batteries.  They tend to hold a working level voltage until they are almost fully discharged, then they just drop out.   So, in this situation, I personally would only consider a LiPo battery.  You will get more run time that a similar size lead acid and they are a whole lot lighter.  

If the boat is not up on plane to the point the hull spray is not back to or behind the drivers seat and properly trimmed out, the hull torque is usually going to be too much for the trim tab under the anti-cav plate to do any good.  It usually works when you are up at a good cruising speed with the motor trimmed out, that's where you adjust it for neutral steering.
Coming off plane or slowing way down the increased drag on the hull is going to make the boat want to turn right and there is not a big enough trim tab made to compensate for that.
 
One other note. If the motor is on a jack plate and you have it raise up a lot, you usually have to put a trim tab on the skeg because the one under the anti-cav plate is not in enough water to do anything.

If I had to guess, the model number is J8RCEC which was made in 1989.

There is one flaw in your math.  A 105 Amp hour (Ah) battery is only rated at 105 Amp hour with about a 5 amp draw on it.  They use the 20 hour rating.  So, if you only drew 5 amps off the battery you could use your math.  Not very realistic.  
As the current draw increases, the Ah capacity decreases.  The exact formula for that, I don't have, it can vary somewhat with the type battery, but it is a rather significant reduction.  At 33 amps, it could be as much 25%.  
If you use the Reserve Minutes (RM) rating, that will get you a little closer to a realistic number.  The Duracell batteries are made by EastPenn and they use a 23 amp rating, some companies use a 25 amp.  What that means is a fully charged battery will run the RM time with a 23 amp draw before it's fully discharged.  Now, since it's harmful to most batteries to fully discharge them, you need to reduce that RM time by about 20 - 25%, the leave some charge in the battery.  By then, you are going to be moving so slow, they are almost useless anyway.  
Then you have to figure if you are going to be drawing 33 amps, then you have to reduce it another 10 or 15% 
So, what you have is some batteries that are closer to about 75Ah instead of 105Ah if you plan on running them at a 33 amp draw, and then leaving at least a 20% charge in them, you find your run time is a whole lot less than your math says it is.  
You actually went the wrong direction when you went with AGM's if you wanted max run time.  An AGM, group 31 is a 105Ah, a group 31 in a good flooded cell is about 130Ah. 
The lead acid battery that's going to give you the absolute most run time, is the Trojan SCS225 but you are not going to run down to Sam's Club and buy one of those for $175.  There's not a lead acid battery on the planet in a similar size battery that's going to beat it.   
 
I just noticed the chart you had with speed and amp is for a 80# 24V.  If your 36V 112# motor is 50 amps, that's a huge load and you can reduce my numbers even more

I'm looking for a new 7" LCD with Chart, down scan and side scan, structure scan etc.
Looking at the Lowrance but they are not real clear on exactly what you are getting in the deal.  The HDS gen 3 a Ti and some other but not real clear if they come with everything for the listed price to provide all the features the advertise.   Things like the Structure scan and the transducers.  I see one list a 3 in 1 transducer but not real clear if that's just a band aid to make you thing you are getting something.
I really don't want to spend $1,200 - $1,500 for a package just to find out I need to spend and large chunk of money for some special over priced transducer or additional unit to really get all the features I want.
I have the old Gen 1 HDS units with the Structure scan that lets them all network together, but looking to upgrade and only want one unit so don't need to networking, but to want all the Structure scan capabilities, 
 
I have always used Lowrance for the past 50 years but I'm not locked in on Lowrance.  I've heard some bragging about the Garmin 7" and others on the Hummingbird.  Just looking for a unit that has a really good side scan/structure scan.  I'm sure they are all made in China now, so it's just a matter of whose technology you want to use. 

Do you plug your charger directly into a drop cord or do you have a plug in the boat wired to the charger you plug the drop cord into?
Did you plug something 120VAC into the drop cord and make sure it's good and have power to that point? Things happen. When I first went in the Air Force I had a Staff Sargent training me on a radio, he had spent 20 minutes tearing it apart because it had no power output when I looked at the back of it, handed him the plug and asked if that needed to be plugged in first.
If you have a plug wired into the boat, check the back side of it and make sure no wires got knocked loose from it during the detailing and moving everything around.

I have built a couple of wooden boats and helped two friends build those very nice wooden canoes. While they are very nice looking and handle great, they are too heavy for my likings.

Since there was no screw head in there with it, most likely it was a peg and the end was braded to hold the plate.  It looks easy enough to me just to level it off, drill and tap a hole in it's place and use a screw to hold it.  Just make sure of what you are drilling through and into.  Going all the way through is not going to hurt a thing, provided the drill bit doesn't hit something but there is no gas to leak there because the float level is well below that point and you can put a drop of locktite on the screw to hold it and seal it. 
 
Distributor screw kits are a good source for small screws like that. 
 
The carb bowl vent hole is probable up inside that section and that's nothing more than a splash guard to keep gas from sloshing out the vent hole in rough water.
 
Actually, you might be able to put the plate back in place, using a small pointed punch with about a 45 degree angle on the point in the center of the peg and expand it out enough to hold it, but the sure fire way will  be to use a screw.  That is fairly soft metal and will expand fairly easily.