I don't think Emperors mind affairs as long as things are made clear. Lots of marriages are arranged and not done out of love, so I don't think it's extraordinary for the Empire to have had several Emperors and Empresses with known affairs. This is also putting aside the possibility of polyamory.
I mean, obviously it's the type of things people in court titter and gossip about, but it's not the kind of thing that can cause widespread trouble considering how easily Heidi dismisses it.
I suspect it could cause widespread trouble, if it were ever actually proven to have a basis beyond gossip. It's the sort of thing society turns a blind eye to, as long as it doesn't become undeniable. Or as long as the monarch isn't looking for an excuse. And historically speaking, it's generally more accepted for men than women. As long as it's not homosexual, because then it's a crime against god and you get in lots of trouble (again, historically speaking rather than personal views).
As is, because Mathilde and Heidi aren't actually doing anything, the worst it can do is be scandalous without being problematic.
Winter is terrible time to for war, even more so when you are the one attacking.
That just means the defender has had all the spring, summer and at least part of fall to ready their defenses and gather food in preparation for siege.
And as mentioned, generals are a thing.
Winter not being the war season has less to do with enemy food supplies, and more to do with the ungodly logistical difficulties (forage and transport are very difficult) and the fact that seiging an enemy fortress is absolute hell in winter, because half your army will freeze, die of horrible diseases, or starve.
[X] Ulric
Choosing Ulric (or another popular Empire god) would be the kind of surprising smash hit that I think would be really successful. An emperor genuinely blessed by a god would have a pretty great leg up on the competition. The other candidates for Emperor would rather obviously not be blessed by a god and thus obviously inferior candidates.
Sure it means less direct devotion for Ranald but this would still be pretty good. Also I kind of like the idea of giving out such out of left field advice. It feels very Mathilde to suggest something like that.
Wizard would be my close second choice though.
Ulric is somewhat contentious. Lots of tensions between Ulricans and Sigmarites, and Reikland is the heart of the Sigmarite faith. Also, being blessed by the gods is not a thing people just work out when they see you. Magnus had to have pretty much every major god send a direct sign that they favoured him to convince everyone. Finally, Sigmar would be a genuinely better choice politically, because he's the god of the Empire and was Magnus' primary deity, so there's good associations with Emperors blessed by Sigmar.
We haven't seen any particular anti-Grey bias (besides general anti-Wizard prejudice) from either the Emperor or the Elector Counts we've interacted with with because at their level secrets and spying is normal and they don't really worry about getting arrested because they defrauded the treasury for a couple thousand gold.
To be fair, Mathilde has spoken to what, three Elector Counts? Abelhelm and Roswita were and are more Witch Hunters than traditional Elector Counts and the last was Luitpold, who works with Wizards semi-frequently and theoretically has no accountability.