Delicious variations abound, so it is hard to make absolute statements of what is "officially" a ladyfinger. In my pastry class, we were taught that the classic recipe for ladyfingers is close, but not identical, to genoise. Store-bought ladyfingers can be anything from, essentially, a meringue to a shortbread cookie.
Foam-based batters -- i.e., those that do not use chemical leveners or yeast -- can be divided into two categories: whole-egg and separated-egg. The main difference in results is that the latter tend to be stiffer and can better tolerate being overcooked a little. The primary example of whole-egg batter is the genoise cake. Because the whites are whipped together with the yolks (and sometimes extra yolks are added), it is a finicky batter, requiring more skill to assemble properly than others, but when it works, it is more tender. Separated-egg batters are far more common, as they are easier to assemble; the key is that you whip the whites into medium-to-stiff peaks, then fold it back into the rest of the batter. Examples include chiffon, dacquoise, and the French "biscuit" cakes.
Most recipes for ladyfingers use a separated-egg batter; others use a simple meringue. The meringue version will make a crisp cookie, the version with with the yolk, soft. Either will work well for uses like tiramisu or trifle.
Here is a recipe for ladyfingers from Allrecipes.com and a recipe for genoise from Epicurious. For my ladyfingers, I add a pinch of salt and some flavorings, such as vanilla or nut extract, but the base recipe is essentially the same.
It could either be a problem of heat transfer or leavening.
Heat Transfer
Probably the bottom "jelly" potion of the cake is not getting enough heat. I can imagine some possibile causes.
- Ovens that are not fan assisted are generally warmer at the top than at the bottom. Generally you should have the cake in the middle of the oven, but for some ovens you may need to move the cake higher.
- Plastic bakeware (or silicone if that's what you have) typically conducts heat less efficiently than glass. And glass has poorer heat transfer than metal, so try using a metal cake form.
- The oven temperature might be wrong; oven dials are often wrong by a few degrees (I once had an oven that was out by ten degrees C). Get an oven thermometer and put it on the same shelf as the cake.
Another possibility is that you just need to leave the cake in longer, but from what you say it sounds like the upper portion of the cake would then be overcooked and dry.
Leavening
In your ingredient list you didn't mention a leavening agent. That could be a problem in itself if you are expecting air bubbles. Typically you would use baking soda or baking powder, or a combination of those two (depending on the acidity of the batter) to introduce air.
It is possible to introduce air without a chemical leavening agent by, for example, folding whipped egg whites carefully into the batter.
In either case, air does not "flow into the cake". It needs to be introduced either before baking or by a leavening agent during baking.
Best Answer
There are several major types of mousse, made from different bases, and with different flavor elements.
Depending on which one you are using, they may have varying requirements. As MandoMando mentions in the comments, assuming you are using a mousse based on whipped cream or whipped egg whites or a freezer-stable thickener (neither gelatin, agar agar, nor carageenan are freeze-stable), you can simply freeze the cake for transport. This tends to work very well, although you may have issues with condensation when thawing it, so you want it very well wrapped (usually after freezing, so as not to muss the icing).
Chocolate mousses are often fairly stable due to the cocoa butter, which is solid at room temperature; some can also be gelatin stablized. See the linked recipe in
Chocolate mousse - methods to have firm foamy mousse using white egg and cream
for inspiration, where the confection is built with two layers of mousse, one which is stablized with gelatin.
Many fruit mousses tend to be made with gelatin, so increasing the quantity of gelatin will give you a stronger mousse.
See also: Hydro-colloid primer for perhaps more information on hydro-colloids than you ever wished to know.