It's been long enough to be caught up with Dishonored 2 for the most part... So, what is it with the Kaldwins and getting it on in taboo relationships and ending up with kids who really should not be eligible for the throne ever-- (okay but c'mon we should start referring to Corvo as "the help")
Spoilers for Dishonored 2 to follow
Idk… the Kaldwins are like, overall good (neutral good? maybe) but have some small character flaws. Like they try to be good to their citizens but they have one or two things going on in the background that aren’t 100% legit. My general impression is that Euhorn and Jessamine tried to be good to their people, especially not overtaxing them, making sure the poorest weren’t living in absolutely terrible conditions, etc.
Specifically, Euhorn Kaldwin - if he is indeed Delilah’s father, and it’s presented to us in the game that apparently that is where Delilah is basing on claim on the throne, and that it’s not clearly 100% accurate BUT we have no other evidence presented to the contrary - it would seem that he was for the most part a wise and gracious ruler, and it was an indiscretion on his part and then keeping Delilah around and letting her play with Jessamine without acknowledging her as a natural daugher OR treating like a princess as he actually said he would (again, according to Delilah) that led Delilah to begin her plotting. I’m not saying that separating Delilah from Jessamine entirely would have been the best thing to do, nor am I suggesting that separating them would have prevented Delilah from turning into the amazingly crafty, talented and motivated individual that she is in Dishonored 2, but according to Delilah’s recounting of events, it seems contributory to her antagonism.
We aren’t shown Jessamine’s entire childhood explicitly and in great detail, but it seems like she had few friends, such that Delilah ended up actually being close enough to feel betrayed when Jessamine (apparently) blamed breaking a precious item on her and she was whipped (and her mother consequently dismissed from the kitchens).
As far as we know, it isn’t necessary in the world of Dishonored for the heir to the throne to be the child of the Empress/Emperor AND their spouse, rather, merely that the heir is the child of the Empress/Emperor. It seems like Jessamine and Corvo’s affair was common knowledge (a ‘badly-kept secret’ in the word of in-game texts) and, up til the start of Dishonored 2, any issues with Emily’s rule didn’t seem to stem from her being born illegitimately, and from a non-noble father.
To reduce the issue down to its most basic, it seems like Theodanis Abele’s gift of Corvo (the youngest person to win the Blade Verbena I think) to Euhorn Kaldwin was a political gesture and that is basically where the whole issue of Emily and her parentage stems from. It’s all political. In the first game, we’re not really told that much about how the ruler of the Empire of the Isles actually needs to maintain peace and balance between four kingdoms - Tyvia, Morley, Gristol and Serkonos - we actually only really learn this in Dishonored 2, where more emphasis is made, with the map of the whole Empire with its distinct capitals. The neatest comparison that comes to mind is Westeros in A Song of Ice and Fire, where the King (or Queen) who sits on the Iron Throne rules with a single mind, but rules over seven kingdoms which are united.
The fact that Corvo was around to fall in love with Jessamine is based on a political decision to attempt to more strongly bind together Serkonos and Gristol, and while it certainly didn’t help that there were apparently no other good suitors who Jessamine liked enough to marry, Corvo being around originates from the political history of the Isles.
I’m not sure I really know enough to talk about Delilah or Emily being ineligible for the throne. Delilah - well from Dishonored 2 she’s not exactly a benevolent ruler, so I guess we can say that she isn’t appropriate to rule the Empire. Emily is certainly well-educated, trained in the Royal practices and also experienced in dealing with assasination/kidnapping attempts; but it seems like she has a yearning to climb rooftops and explore the Empire not as a ruler, but a citizen, which when related in-game, is given as the reason for her rule being not 100% perfect but not necessarily actually being a good reason for that state of things. This relates to something Corvo says when he turns up back in Dunwall in the last mission, along the lines of “being born royal doesn’t give you the right to rule, you earn it every day by serving the people”, and this relates to power and the abuse of power, which is one of the themes of Dishonored 1 & 2. I don’t really have any concrete answers for your question, sorry - all I can say is that this is one of the things the developers have stated that they are interested in exploring, through the use of the two games.
I hope that answered your question?