How do you suppose these sons would be raised and educated? Frederick William IV was a romantic, but I'm not sure how this would carry over into their upbringing. I have no idea what sort of person Elisabeth was beyond supposedly being a loving and supportive wife, and someone with at least some interest in theology, so no real clues there either.
If Joachim is given a standard Prussian military upbringing, he's likely to run into many of the same conflicts with liberal politicians as William I did IOTL. While a regency likely won't be required at any point, I could still see William being a major influence in king Joachim's reign if he has the same Prussian mindset.
As for marriages; we won't necessarily have to only look in Germany for potential spouses. There are many legible Russian grand duchesses available, for instance. They mainly married other German nobles OTL, but with three Prussian princes, they might as well try to maintain dynastic ties to Russia as well rather than only marrying Wettins and Wittelsbachs.