Perhaps a simple example of how a float bowl works is go and pull the cover off your toilet, flush it and watch what happens in the tank; your carb float bowl operates exactly the same way.
It seems like we are talking about several things simultaneously; spark, fuel, and timing. While they all are symbiotic they must be diagnosed on their own merits. Your comment about it starts initially but its a crap shoot later provides some insight to your mind set. First let me say that the comments by Jimy and myself about each car having its own procedure to start remains very true, however it extends to warm starting as well. In my experience, many carbureted cars act very different from cold to warm/hot. Most of the cars I have dealt with require not pumping or throttle once they are warm to start (or very little). In your case, you need to do some experimentation to determine what you need. When it is warm, you might try giving it 1/2 a pedal to allow some vacuum from the turning over engine to draw some fuel in.
Since you are "learning" let me explain this issue (if you know this I apologize in advance); a engine is a big air pump and while turning over (the pistons going up and down) it creates vacuum. This vacuum pulls fuel and air down through'from the carb and draws it into the cylinders where it can be ignited by the spark plugs thus creating power (i.e. running). Therefore, the engine must be able to receive both air and fuel in this sequence to operate (the spark must be happening in the correct timing to facilitate all this as well). All that said, each of these systems must not only be operational and functioning correctly but also set up to work with one another. Arguably one of the most important aspects of this specific situation is timing. You can be a several degrees off timing and the engine will run but will start hard (assuming all of the other related components are doing their jobs correctly).
Now, having said this I am not advocating you start twisting the distributor to alter the timing, rather I am suggesting that you spend some time trying to figure out what you engine wants in order to run in each condition. Right now you have the weather fighting you; when that car sits a day in 20 degree weather it is actually harder to get started because the cylinders and heads are cold and are actually combating the spark however it does assist the fuel making it cold and therefore dense.
So, what I would recommend is that you go out to the vehicle when it is cold. Since you seem to have the cold start somewhat figured out, start it and let it warm up. Once warm, shut it off and let it sit a few minutes; try and restart it but only for a few seconds. If it starts turn it off and let it sit longer (say 10 minutes) and try it again. If it doesn't start immediately, stop and let it sit a few more minutes. Now, go in and depress the pedal half way and try to start it. If it starts then you are on the right path. If it doesnt, let it sit for a little while and go and try again but this time hold the pedal 3/4 down. Again if it starts you moving in the right direction, if not repeat this except give it one pump. I think you can see where this is going and the idea is to determine what your engine wants when it is warm. Understand this though; older carbureted cars typically do not start like modern EFI cars; they will turn over a little while before catching. Even when these cars are outfitted with EFI they still tend to turn over a few times before firing up. This is a result of engine design and inefficiency of the design; the air fuel has to travel a ways to get into the cylinder in sufficient levels to create an explosion large enough to run.
It very well could be that your engine is as good as it is going to get without vast sums of money and some significant aftermarket equipment (i.e. different or better ignition system, better intake, better carb or EFI, better exhaust, aluminum heads, etc.).