How to make the sauce that is underneath Flan

A medium saucepan would probably be in the 8-10 inch range.

Your sugar isn't colouring because the heat isn't high enough or it's high enough but not long enough. Just keep cooking, the change will happen. You can try less water but if you're not familiar with working with sugar, better to err on the side of caution; I do work with caramels a lot, and when I know I'm liable to get distracted by other things I have on the go I purposefully use more water than needed to give myself breathing room.

Are you using the correct size of dish as specified by the recipe? Some of the caramel will dissolve into the filling as you pour it in and as it cooks; I wouldn't worry about the thickness at this stage.

Beyond that, I would personally stay far, far away from anything done by RR. I don't criticise cooks lightly, and I generally try to cook some of their recipes before I'll pass judgement. The recipes of hers that I have tried, both from 30 minute meals and the website, are uniformly awful. Poorly written, ingredient quantities are off, and generally they just taste nasty.

Soft melt increases with fat and emulsifiers. Emulsifiers make the mouth feeling smooth and silky, but not as rich as fat. Firmness increases with proteins. Thickness increases with dry matter (a bit), proteins and additional binding agents.

I don't know which feeling you want to achieve, but here is a list of your relevant factors and what they do. The info (plus the info above) should be enough to tweak an existing recipe in the direction you want.

• ratio of eggs to milk: more eggs mean more dry matter, more fat, more emulsifiers.
• egg yolk to egg white ratio: egg yolks have fat, emulsifiers and protein. Egg whites have protein only.
• condensed, evaporated, regular milk or cream. Let's take regular milk as base. In comparison, evaporated milk has proportionally more dry matter and fat. Condensed milk has even more dry matter because of the sugar. Cream increases the fat content a lot, and because fat is dry matter, it also increases dry matter - but the ratio of fat to overall dry matter is much higher than in milk (regular, evaporated or condensed). Don't count on the proteins of milk much, they are a relatively low amount and have been overcooked anyway (unless you use raw milk or cream).
• Binding agent. Flour and starch add thickness only (flour has some protein too, but not much). Tapioca, arrowroot, etc. are starches. Gelatine is a protein, adds both thickness and firmness. Gums add thickness and emulsify. If you use something more exotic, ask in a comment.

The rest of your list doesn't really matter. Or, more accurately, it does, but there is one correct combination and you shouldn't tweak it in attempts to change the mouth feel.

It is actually wrong to measure cooking time. What you want to measure is the temperature your custard reached. Yolk emulsifiers start to work at about 50°C. Starch needs 70°C. The first types egg proteins start binding in the high-70s. Somewhere in the high 80s, other types of egg proteins bind too. You don't want this last binding to happen - if it does, your custard gets too firm and exudes some liquid after cooling. If you take it even higher, it gets not only firm, but grainy. If you let it simmer or boil, you also get biggish liquid-filled bubbles in a baked custard, and a sandy texture in a stovetop-stirred one.

So, the correct temperature is in the low to mid-80s. I usually aim for 85°C. The correct time is whatever it takes to reach that temperature, and depends on the rest of things you listed - but the relationship is so complex that the custards you'll ruin to determine the right combination for your oven/stove and dishes by trial and error will cost more than a candy thermometer (mine cost under 15 Euros). And if you go with the trial-and-error method, you won't be able to achieve a predictably good custard outside of your own kitchen.