Remove this HVAC damper

I don't see a problem with that, other than the existing duct doesn't look wide enough to fit both. You'll have to get/build a reducer that goes from something slightly wider (that can support both circular ducts coming out of it) to the existing square duct size. Most metal shops can build this sort of thing for you - I had a couple 45-degree elbows built to re-route some ducts in my basement, it cost me around $50 I think.

 | |   | |
 | |   | |    <--  existing two round ducts (one moved)
 | |   | |
+---------+
|        /
|       /    <-- new reducer
+------+
|      |
|      |     <-- existing square ducts
|      |

The main thing is to keep the number of elbows to a minimum. In fact, in your case it may work better as it looks like you'll reduce the number of elbows by one.

Another option would be to extend the main duct, and split the two round ducts off at the end. Not sure how long those are now, but the box stores carry a few regular square duct sizes you can buy and assemble/install yourself.

In theory this is what HVAC professionals are supposed to do. In practice, most of them rely on antiquated rules of thumb to size equipment and use existing ductwork, even if it has major problems. A certified energy rater is who you're looking for and who will be able to definitively tell you what you need to know, but you can learn a lot yourself to be a more informed consumer.

The correct size of equipment for your home is discovered by doing a Manual J load calculation. You can do an 85% as good approximation yourself in 15 minutes here: http://www.loadcalc.net

For a 1930s house, unless major air sealing, insulation, and window replacement work has taken place, the load may be as high as 1 ton of AC per 500 square feet and 30 heating BTUs per square foot. But those would be worst-case scenario numbers; run them yourself with that web tool and get a ballpark estimate of your house's loads.

The most appropriate type of equipment for your house depends on your climate, your budget, and the fuels available. If you live in a mostly-cold climate and have natural gas, then a gas furnace makes the most sense. If you live in a mixed hot-and-cold climate but it doesn't get too cold, then a conventional heat pump may be a good bet. If it gets both very hot and very cold, then AC + a gas furnace may make the most sense if you have gas; if you don't, then a Japanese ducted mini-split or conventional heat pump is a better choice.

As for your existing ducts, it's impossible to say whether they're adequate or not without knowing anything about them. But they may be inadequate if they are in the attic, or if they are in a vented crawlspace, or if any rooms get much hotter or colder and other ones, or if a lot of air leaks out of them.

Another thing to keep in mind is that you're not really hiring someone to design your home HVAC system; your home already has an HVAC system (well, the "H" and "AC" parts, at least). If every part of it is unsalvageably wrong, expect to pay mucho dinero to completely replace everything. New equipment and new ductwork and all the demolition and drywall repair entailed by such a job will probably cost you $15k or more... possibly much more if you live in a high-labor-cost area. Consider how much you actually need to replace and how much you can simply live with, especially considering that many comfort-related deficiencies in a house's heating and cooling infrastructure can be ameliorated by improving its insulation and air sealing. There are many such opportunities in an old house, including the extremely low-hanging fruit of air-sealing the basement, weatherstripping any old windows, and adding more attic floor insulation.

In general, it's a good idea to do all this work before replacing your equipment, since if you buy correctly-sized units now, they will be too big after you perform the work. You can mitigate this to a certain extent by purchasing "modulating," "variable speed," or "two-stage" units, as they are capable of reducing their output at low-load conditions. In theory you won't have to use your furnace for another 4+ months at least, so you have some time to tackle those sorts of projects before having to replace it, which will allow you to purchase a smaller one and save money, or a better one for the same price.