How to improve the texture of frozen yogurt?
First, you shouldn't have the expectations that your ice cream (or frozen yogurt, or whatever) will turn out to be like storebought. You can get to a "better than now" state, but not to a "out of the heart brand tub" state. 

For your ideas, some are off the mark, some are worth trying. 1) a hydrocolloid is generally good in ice cream, since it binds water and prevents it from making large crystals (helps with "icy"). Pectin is an unusual choice though, guar gum works better. But you can start with pectin and then graduate if not happy. 2) don't reduce the alcohol, it will get icier. 3) for the purposes of setting, golden syrup is sugar. You can add more if you want, substitution won't help. 4) frozen yogurts are not custards, cooked yogurt is not tasty. 5) yes, reducing the available water can make it less icy, so do strain. 

When you add the emulsifier, you will have done about as much as you can at home to get rid of the crystals, seeing that you are also already adding both salt and alcohol. So, not much more that can be done in that department, except of course proper churning. 

And here we are coming to the crux of your problem - you are not churning properly. The "blend, then reblend" method is a substitute that never works as good as an ice cream machine. Also, you are mentioning a "stick mixer" - so you most likely don't have a powerful blender which can turn a block of water ice into snow, just a standard model. You probably have zero overrun, so of course it is melting right away. 

The solution is to buy an ice cream machine. Make sure to read test reports, there are some disappointing models on the market. A standard frozen-bowl style will already be an improvement, although it will still have some melty tendency. If you want gelateria-style ice cream and have the space and the money, you'll have to invest in a compressor machine. I doubt that you can ever get supermarket-style at home, unless you get into serious equipment and recipe modding.