I just wanted to say that in my experience, a 3/16" bit is bit small but a 1/4" bit is definitely rigid enough to cut well on a 10" lathe, especially if it is ground well. I use 3/8" bits almost exclusively on my 11" lathe because it allows me to stick the tool out however far I need to get the job done and it fits the AXA tool holders better. 1/2" is too big and doesn't confer any advantage to me. Bottom line is a 1/4" or 3/8" bit will be far more rigid than your lathe is, and either will allow you to get them on center height.
The parting tool is shop ground and will work fine for smaller diameter work if it is sharp. A P-type blade will be better, especially if you go with a P-1N or P-1.
The tool blanks you got are good ones - Morse, Mo-Max cobalt - both top of the line. The brazed bit will be okay but needs to be sharpened well to work if you can get it on center.
You are going to find that the lighter the lathe, the more important it becomes to grind good tools and get them dead on center height. Bigger, more rigid lathes have more latitude but small ones need every advantage they can get.