BLOOD AND THUNDER!
Today's Yesterday's Noire's B-day (and I might as well throw in The Avatar's B-day here as well since ACC got a B-day for his and it's as good of a day as any).
Noire, like Owain, Inigo, and Cynthia, cannot inherit the full 9 classes that the other kids can as their class sets come with too much overlap (or in Cynthia's case, none of her non Avatar options do her a lot of good). That said, Noire preferred father is Gaius due to getting her on the Galeforce and the fact that only Nah and Kjelle really want him otherwise (and Kjelle would rather have Donnel first). Gregor, Donnel, and Lon'qu are also good fallbacks for her but Gaius is pretty much the best in terms of skills offering Galeforce, Lucky Seven, Astra, and an inheritance of either Sol, Counter, or Axebreaker. Counter is actually a pretty amusing choice for Noire simply due to her default class being Archer, but like most children, she doesn't really care about her default class (that said, Counter is still pretty useful for Double Duel), but she'd probably want Sol instead since Axebreaker is almost as useless as Bowbreaker is on a non-flier. Her mother Tharja gives her an interesting advantage in that she can pass down Tomebreaker to Noire (if you bothered to level her up for it and not reclass her to something else or bench her entirely), which when paired with Bowknight's Bowbreaker, can enable Noire to take Dark Flier and pick up Galeforce with almost no effective damage to worry about (only Beastkillers and Violent Axes can do effective damage and the latter doesn't really show up at all in enemy hands). This means that Gaius!Noire can have an amazingly solid start and just in time for the main game's endgame.
As for stats, Noire also has an interesting start in that she's in the one class that can fix the biggest problem she inherits from her mother, a lackluster Skill stat. Moreover if Tharja shacked up with Ricken instead, Noire would have the highest possible Magic modifier outside of a Magic specced Morgan (too bad for the Archer tree + Great Knight class overlap, at least she'd get Sage and Paladin out of the deal). Of course Noire will inherit the worst Skill and Luck stat modifiers out of all the kids without help and a Gaius sired Noire will have +1 to Skill and -4 to her Luck modifier while Donnel!Noire will have -1 Skill and +1 Luck. With a properly specced Avatar, Noire!Morgan could wind up having either +10 to Magic, -4 to Skill, or -7 to Luck.
The tl;dr for Noire is that she's pretty self sufficient on her own and hard to fuck up, but with a good combination she can become one of the most solid characters in the game. She really wants to take to the skies though.
As for The Avatar him/herself, aside from running errands for Lord British, the Avatar can basically be anything and do anything (except for Morgan and the other child born from the Avatar's union for obvious reasons). The Avatar can choose to diversify his/her stats betwixt many of the major stats (if the HP build is taken) or to specialize in one particular stat and minoring in two others (every other option), and likewise can the Avatar take a massive hit to stats based on the penalty taken. With the specialty the Avatar can gain a +4 in a particular stat, the only characters who have a modifier that high are children and a few of the post game characters but with the harshest penalty of -3 can Maribelle, Tharja, and certain children with specific parentage can match. No matter the build though, the Avatar will always have a +3 overall stat modifier which is great for a 1st genner but not excellent.
For skills and class access, it'd be easier to name the ones the Avatar doesn't get. The Avatar will always have access to the Tactician and its promotion Grandmaster as well as the Cavalier, Knight, Myrmidon, Mercenary, Archer, Thief, Wyvern Knight, Mage, Dark Mage, and either Priest/Cleric class trees as well as either the Fighter and Barbarian (and Dread Fighter if you include the DLC) class trees or the Pegasus Knight and Troubadour (and Bride if you include the DLC) class trees, and no matter what the Avatar cannot access the the Lord or its promotion Great Lord classes (although Morgan cannot access Lord/Great Lord even if he's Chrom's son but Avatar!Lucina still can), the Villager, Taguel, or Manakete classes, nor can s/he or their progeny can access the Dancer, Overlord, and Lodestar classes. Either way that's 54 skills + either 8 or 10 skills (depending on gender) + up to an extra 6 from DLC, that's anywhere from 62 to 70 skills and a whole shitload of combinations available to the Avatar.
And while Morgan is pretty much straight up superior to the Avatar in class/skill access and stat modifiers, and the DLC/Spotpass characters are basically premade Avatars, the Avatar will always maintain one advantage over these pretenders, and that is the ability to support EVERYBODY (outside of DLC/Spotpass characters; not to be confused with the postgame characters that are unique and aren't just Avatar copies). Regardless of gender, the Avatar can support anybody and everybody, including the characters with only one support option (which is, of course, to the Avatar), and moreover can achieve S rank (and thus marry) anybody of the opposite gender (progeny aside for again, obvious reasons). While this doesn't mean much to the Avatar him/herself, it could mean everything to Morgan who can inherit classes or skills otherwise exclusive to certain characters or locked to their progeny as well as inheriting the boosted stat modifiers that 2nd genners get (as 2nd genners' modifiers are the sum of their two parents and an overall +1 boost to every cap, Morgan's modifiers will be the sum of three parents worth of modifiers plus that overall +1 boost; which unfortunately does not get repeated) netting Morgan modifiers of up to +11 in some stats and/or an overall stat modifier total of +18.
As for in-game performance, the Avatar will immediately hold three advantages over everybody else. First is availability, matched only by Chrom. Second is Veteran providing an accelerated experience growth rate, provided s/he remains paired up with somebody. And finally is weapon advantage holding both means of attacking Defense and Resistance at the same time, as well as a built in ranged option that doesn't deny him/her the enemy phase, plus a means of taking advantage of early game weapon triangle which is usually (although not so much in this case) heavily populated with axe users. Moreover given the Avatar's mission critical status, one could think of the Avatar as a sudo-Lord character with a sudo-exclusive class (as long as you don't look online), however, the Avatar is not a lord and is thus not mandated in any fight save for prologues 1 & 2, and chapters 1, 2, and 23, and thus can be perma-benched. Additionally, the Avatar's overall growth rates kinda suck (made up for the fact that s/he'll have a lot of levels to go through to make use of their nigh unfettered access to the reclass system).
The tl;dr is that the Avatar is more or less the main character and has nigh unlimited options available to him/her at any given time. Once you get to level 10, the doors will fly open the world will become the Avatar's oyster.