Is the eldest child of the monarch favoured over younger ones?
Not one Scarlet so far has been a child of a previous one. After the First Scarlet lost two children she tried to train to the Manse, there is an unspoken rule that one does not train one's own children. That is something that Princess Nemone must contest against - the expectation that no child of a Scarlet would be able to do it properly given precedent.
Is the bias towards females codified, coincidence, a societal bias, or an emerging but unofficial tradition?
Well, it's just that women are more rational, sensible, logical and less ruled by their hearts than men. Men are like fleeting pretty butterflies, torn between their loyalty of their House of birth and the House they marry into, and that leads them to be better suited for artistry and the more ephemeral pursuits rather than the more solid, grounded, logical thought of women. Men live in the knowledge that the blessings of the dragons pass down less strongly through them and the aspect of the mother more strongly determines the aspect of any child than their own aspect - they're just not quite as in tune with their element as women. And of course, the Houses don't tend to push their men as hard - after all, it'd be quite a waste if someone you'd invested so heavily in and were weighing all your hopes on went and married someone from another House and now had split loyalties!
Now, that's not to say that men can't excel in whatever field they choose to pursue. But it's just a question of bell curves. Have you noticed that all the best sorcerers are female? It's just women are more talented at the rational thought that Devonian sorcery requires, and men in their emotional sensitivity tend to be better in less... structured environments and tend to study Salinian sorcery, which just isn't as productive at producing research papers at the high end. There is a standing issue, and the Heptagram does need to work better at getting more men into studying more academically reputable sorcery, but that's a question of culture and, yes, male biology.
Evidently the current monarch select their successor (like the early Roman emperors?), but has the procedure been put into law, or is it fairly ad-hoc, ruled simply by precedent and the current Empresses willingness to force her choice down the Houses throats?
The Empress or Emperor can select whoever the fuck they want and train them however they like.
It's up to the Manse whether it'll accept them. You can't force it. The First Empress made that entirely clear when she told the heads of the Houses that if they wanted a say in her successor, they could go into the manse with her and face the Realm Defence Grid themselves. Four accepted. Three returned, shaken by what they'd seen, and told the others that it had destroyed the one who'd faced it, utterly. The Empress announced that a lot of her candidates were going to die anyway, so her heir would be someone she found who the Grid would accept, nothing less and nothing more.
The heir of the Scarlet is only a true Empress-in-Waiting once they've activated the Grid themselves successfully. That's the ultimate proof, and that's why the Realm is in trouble right now. Princess Nemone is only half trained and hasn't tried for a successful activation yet, so she isn't the Empress-in-Waiting. There is no Empress to complete her training. No one else can do it. She's the foremost candidate, but she knows she doesn't know what she doesn't know. It scares her. She knows so very many ways the Grid could kill her if she doesn't meet its standards.
(The Grid was built by super-genius Solars who exhaustively built it so no lone madman could use it to destroy Creation. It
will kill anyone whose highest goal when activating it isn't the preservation of Creation - and it will refuse to activate for anyone who has defences up which would stop it killing them. It will kill you if you have mental defences up which stop it from reading your intent. It will kill you if you are not one of the types of Exalt which fought in the Primordial War. It will kill you if you have supernatural mental influence on you because that might mean someone is using you as a pawn. It will kill you if you are tainted by the Yozis. It will kill you if you are tainted by the Wyld. It will kill you if you are tainted by necrotic Essence. Even beginning the activation process is effectively an Eclipse oath that you will not prevent it from killing you if it finds you a threat to Creation.
If a lot of Cauldronists towards the end of the High First Age had tried to activate it? Even with the more generous conditions for using it when you're not a lone emergency user like the Scarlets, it would have killed them.
So, yes. If your goal in life is to become emperor, it will kill you. Which means the secret Nemone suspects and Ejava doesn't know is that the Grid will kill anyone who wants to be accepted by the Grid so they can become the Scarlet. The training of the Empress-in-Waiting is hard, painful, and intended to break you so you care more about Creation than your own ambition or even your own life, because that's the only way to successfully activate it as an emergency operator. The Scarlets are all a bit crazy, one and all - but they're a crazy where their towering force of will and sheer heroism (as exceptional Dragonblooded) is devoted towards the good of Creation.)