Girl!Robb would definitely be trained as her father’s heir, there’s
not question about it. Ned and Catelyn might continue to try for a male
heir (and Catelyn would definitely be actively hoping for one that would
secure the inheritance in a way she’d fear her girls couldn’t;
Winterfell has never been ruled by a lady in her own right after all,
much less by a lady with a southron mother whom she favors in looks –
something that could be spun as a mark of her being an outsider and used
to usurp her if Jon Snow ever thinks to try and claim Winterfell for
his own which is a fear that would be much more acute and prominent in
Catelyn’s mind in this AU). But since Ned is not the kind of guy to
subvert laws of inheritance, there is no conceivable way that he
wouldn’t be already grooming his eldest daughter for ruling in her own
right by the time the royals make their way north, a move that would
certainly be encouraged and even pushed by Catelyn since it would mean
that Ned recognizes her daughter as the rightful heir to Winterfell and
is not entertaining thoughts of having the Stark-looking Jon Snow
legitimized and declared as his heir.

As is the custom with heir education, girl!Robb (let’s call her Robin) would be expected to shadow her
father, attend him while he holds court or metes out justice and
accompany him on his travels to his bannermen which simultaneously
teaches her how to govern her principality and familiarizes her to her
future bannermen ensuring their acceptance of her eventual rule.

I don’t think Ned would take additional steps to ensure his daughter’s rule beyond involving her in his rule.

Succession laws mean that she is the uncontested heir and Ned would
hardly try to subvert the law or name another as his heir, not that
there is another to be named heir anyway with Benjen in the
Night’s Watch and Jon a bastard. Even if there are some in the North who
are discomfited at having a future Lady Paramount, tough luck. And it’s
not like Rbbin would be the first Lady Paramount in Westeros as well;
Jeyne Arryn is an excellent example of a Lady Paramount who ruled her
domain capably
and protected the interests of her vassals (from a Targaryen prince, no
less), and the several ruling Princesses of Dorne should not also be ignored. The Maid of the Vale also sent troops to war on the side of
Rhaenyra Targaryen during the Dance of the Dragons so I don’t see why
the North wouldn’t follow their acting lady to war when it’s because
their lord had been thrown in the black cells and accused of treason.
There would be rumblings about who gets the command since Robin wouldn’t
be able to lead the charge herself (since I don’t imagine that even her
status as heir would encourage Ned to have her trained at arms) and the
Greatjon would still challenge her in Winterfell but I don’t see a
reason for the outcome to be different.

(Interestingly, what might
be different is the relationship between Rbbin and Theon and that’s a
divergence that could change the war narrative if she doesn’t have the
implicit trust Robb had in Theon in OTL that made him send Theon to the
Iron Islands, ultimately leading to Winterfell falling and Roose Bolton
smelling blood in the water.)

Re the betrothal to Joffrey, well,
Robin might very well be already betrothed by the time the royal family
shows up. While we know that Ned wasn’t keen on betrothing Sansa
because she was too young, Robin, at 14, would be the appropriate age
for betrothals considering that girls get promised at a younger age than
boys (indeed Lyanna Stark was already betrothed to Robert Baratheon by
the time she was 14 while Catelyn was betrothed to Brandon Stark at 12).
As heiress to Winterfell, Robin would be the greatest marriage prize in the North so
it’s a given that the Northern lords would be vying for her. I’m sure
that Catelyn, mindful of her daughter’s image and keen to empathize her
Northernness, would advocate for her daughter to be betrothed to a son
of one of the prominent Northern families as soon as possible. That
would work to satisfy the Northern lords since it guarantees no outsider
would swoop in to marry their future lady (and potentially rule them in her name, as they would undoubtedly fear) and also win Robin the
support of her intended’s family in case Jon Snow ever thought to press a
claim to Winterfell.

But even if she isn’t betrothed when the
royals arrive, I still lean towards no. Would Robert even pick her in
the first place? While she’d hardly be the first heiress to renounce her
inheritance in favor of the higher station of Queen, there is literally
no reason for him to upend the Stark succession when Ned has younger
daughters that could be betrothed to Joffrey just as easily. On the
Stark side, having the girl who has been groomed for rule and the one
Ned’s bannermen are familiar with abdicate might not be the
best idea. Having Robin give up Winterfell means that Sansa would be
the heiress at a time when Ned is in the South, unable to teach her like
he taught her elder sister or to ensure that his bannermen would
receive her favorably, something that is really essential in Sansa’s
case because she is the most southron of Ned’s children and because at the time, she didn’t even want to remain in the North.
That runs the risk of sowing discontent among the Northern lords with their dislike of the south and risks a backlash to the Starks’ association with the south. With Jon Snow present in Winterfell, a son who is the
spitting image of Ned with his identity firmly grounded in Northern culture, Catelyn would not want the risk of taking her eldest out of the succession in favor of making Sansa heiress to Winterfell. The better dynastic choice would be to betroth Sansa to Joffrey, and Robin to a Northern son.