I'll try to answer this.
Concerning the first recipe you linked, it doesn't matter at all whether you add it first or not, since there is no extraction going on. Theres no fresh coffee, no fresh vanilla. Just extract, which of course is already extracted, so no extended wait time is needed! There might still be room for improvement, letting flavors 'marry' and so, but I'd say a day should be enough.
The second recipe is different. It steeps fresh coffee in alcohol, and here it's going to need time. One month is however an extreme amount of time. Usually, hot water and coffee is steeped for seconds, and cold water and coffee is often steeped for 12 hours. Consider that alcohol is a stronger solvent than water.
But if going for a true alcoholic extraction, pure is better / faster, so you should steep before adding water, and the sugar probably doesn't make much difference.
This said, I'm not sure you should actually let it steep in alcohol at all, and I'm not sure you should dilute it either.
If I was doing this, I would go for water cold extraction. I'd be concerned that alcohol would extract too much, the same as too long hot water steeping leads to bitter, sour, pungent coffee. I'm not sure why you would want to dilute it with alot of water either. Water adds nothing to the party. Dilute it when using it instead, this leaves you free to choose each time. Perhaps you'd like to mix it with milk instead, sometime?
Do you have a french press? Let 1 part coursely ground coffee steep in 3 parts cold water over night. Use the filter / top to filter it, and pour it into your vodka. I'm not completely sure, but I think 1 part extract to 4 parts vodka would be fine. Use the same ratio of sugar as in the recipes you linked, or after taste.
UPDATE:
I didn't completely read the second recipe before I answered. It says to boil the vodka-mixture. Sure, the alcohol doesn't vanish in an hour, but I'd still say it's an unacceptable waste (spirits are expensive where I'm from!). Also, the effect is the inverse of destillation, so you are left with a weaker, but sweeter, drink. I'm not sure why it says to boil it. It says that it's to 'thicken the liqueur', but they could just use less water instead.
Good luck!
Its possible that when you drink a hot liquid, you swish it around your mouth a lot less than when you drink a cold liquid, and thus fewer of the taste buds on your tongue are exposed to it. This wouldn't be surprising: it's hot after all.
Another possibility is that if you're starting with hot coffee, and adding sugar to it, that the sugar isn't fully dissolved when you drink it hot (it takes time, especially if you don't stir it much), but has dissolved by the time the coffee cools. With more dissolved sugar, it is sweeter.
Best Answer
Upon addition of sugar to the superheated coffee, the formation of bubbles (phase transformation) occurs because the fine particles of sugar provide sites for the heterogeneous nucleation of gas from the liquid.
When the introduction of a fine inoculant (such as sugar particles) results in sudden fizzing of a liquid, it is an indication that there has been minor superheating of that substance.
Superheating tends to occur more in microwaves than on stovetops because people use metal saucepans/kettles on the stove, while generally using glass or glazed ceramic containers when heating a volume of water in a microwave oven.
When water is heated in a glass or glazed ceramic container rather than a metal one, the very hard surface of the container means that there are few scratches on it to act as sites for the heterogeneous nucleation of gas. Fewer heterogeneous nucleation sites means less heat loss through the transformation of liquid to gas.
This university site also makes a good point about the tendency for stovetop heating to cause localised superheating in the vicinity of the container walls... However, I think that this would tend to cause more boiling due to localised heating to the point that homogenous nucleation can occur (with the homogeneously nucleated bubbles then acting as further heterogenous nucleations sites) – not because of stirring of the water.
With regard to why your bubbles remained present for the duration of drinking, I would hypothesise that this perhaps has something to do with the oils in the crema of the coffee, and/or reaction of the dissolved sugar to form something that increases the surface tension of the water. (Not sure what you would call the opposite of a surfactant effect).
A final thing I would like to mention is that (as indicated on the UNSW page), microwaving liquids has the potential to result in violent reaction, or even explosion, of the liquid — in other words, you are risking serious burn injuries by taking a shortcut to heat it up.
Adding a powder (like sugar or instant coffee) to superheated water is particularly bad because in doing so, you are introducing millions of nucleation sites at once. That said, you should also be aware that since gas bubbles themselves promote heterogeneous nucleation, simply placing a spoon in the cup can be enough to cause an 'explosion'.
If you really need to microwave a liquid, (e.g., if you are cooking something), consider heating it in a microwave-safe plastic container that has been washed a few times (and is therefore abraded on the inside). You should also stop the microwave to check the temperature of the liquid at regular intervals, rather than nuking the fluid for an excessively long time.