The 250 would do fine, just harder to find and you are going to pay a lot more for it. Converting the 109 is pretty simple, remove old internal parts, install new internal parts and tune up. I would suggest converting the 109 if for no other reason than it makes a good first reg repair/conversion.
You can use pretty much any second stage.
I highly suggest you get the 2 books I list below, they are pretty much the Bibles of the reg tech world. Reg Savvy is a little more on the technical side (the physics of how regs work) and Scuba Regulator Maintenance and Repair leans a little more to the DIY side. The combination of the 2 will give you a really good understanding of scuba regs. I encourage you to make these 2 books your first purchase, they will save you a lot of money and are well worth the cost.
"Regualtor Savvy" by Pete Wolfinger
Scuba Tools
"Scuba Regulator Maintenance and Regair" By Vance Harlow
Airspeed Press Homepage - Books For Serious Divers
Your balanced questions is covered in detail in both books but I will try a short answer. First off, the first and/or second stages may be balanced. It is not a requirement that both stages be balanced, nor does it hurt anything if both are. Each stage handles balancing differently and is not dependent on the other to accomplish the job. The overall purpose of balancing is to keep the second stage performing at the same level (cracking pressure and work of breathing) over the entire range of tank pressures - ~300 to 3000+ psi. This is done by building the first stage in such a way that it's output to the second stage, known as IP (intermedient pressure) stays at a constant (usually somewhere in the 130 to 145 psi range) as the tank pressure changes. With a constant input pressure the second stage operations stay constant. The second stage may also be balanced. This is accomplished by constructing the second stage valves in such a manner that it compensates for varing input (IP) pressure, keeping it's operation constant. I have purposely avoided the details because it gets kinda long winded and the books above do an excellent job of descibing the process in great detail.
All USD/AL Conshelf/Titan/Legend/SEA first stages are balanced. They use the same internal parts in different looking outside packages.
And for what it's worth, balancing is the most misunderstood subject on this board. Read the books, you will be glad you did.