Is it possible to pass every skill check in Long Live the Queen?

It seems that there are skill checks early on in the game which are extremely difficult or impossible to pass because of the mood penalties making it hard to increase the relevant skill.

Is it possible to pass all the skill tests? If it is, how do I optimise Elodie's skill gain in order to do so?

No, it is absolutely not possible to pass every skill check in a single playthrough. There is no single solution to the game, you have to make decisions about what things you wish to pursue and what things are not worth spending time trying to chase.

Every skill check can be passed on its own but only at the expense of failing others.