Probably Constantinople would fall in 1403 or so, and the Ottomans would advance farther and earlier in the East.
Seems likely, apparently the Emperor Manuel II had jumped the walls leaving the city under the command of his nephew to go on a tour of the European royal courts to try and drum up some support for them. If the Ottomans don't have to withdraw the troops laying siege to the city they can probably starve them out. There might be a few draw backs from having a living emperor floating around unlike in our timeline where Constantine XI died defending the city but after he dies off and a bit of time I don't think it would be any worse than the other branches that survived.
The other major event that this would butterfly away would be the Ottoman Interregnum that Bayezid's imprisonment kicked off. Now the Ottoman succession was never a formal affair but the fights for supremacy were never that bad so that misses out ten years of civil war.
Could they reap a substantial part of the Timurid state, maybe as far as Western Persia? Or, as more likely, challenges in the west would force them to stop a significant eastward expansion?
That's a good question. They're established both in the Balkans and Anatolia so they could get dragged in either direction, but if they split their resources they might not do as well as they did in our timeline. They've certainly got some unfinished business with the small Turkmen emirates in Anatolia like Karaman and the White Sheep - it was Bayezid's attempts to annex or bring them under his control that got him into trouble with Tamerlane, he considered them to be under his suzerainty and offered refuge and support to the rulers Bayezid expelled. So I could definitely see him stomping them into the ground and bringing them wholly into the empire, after that I couldn't say which way they were likely to go.
Would the Ottoman state be stronger in the long run, or not?
I tried to search for this topic and I found very little about it.
If I remember correctly it was getting beaten by Tamerlane so decisively that forced the Ottomans to make some major reforms of their military which greatly increased their effectiveness. So on the one hand they've captured Constantinople and probably wont have any troubly running over the Turkmen states in Anatolia, on the other the next serious opponent they meet could be much harder to beat with their unreformed military.
The other thing is would Bayezid be smart enough to institute similar changes to those Mehmed II did when he took the city in our timeline? Not allowing the besieging troops to destroy the fabric of the city, forcing the provinces to send a set number of families to repopulate it, tax incentives to encourage craftsmen and traders, and the major one of appointing the anti-Western Gennadius Scholarius to be Patriarch of Constantinople under his patronage to drive a wedge between the Greeks and the Latins. I honestly don't know enough about Bayezid to say whether he'd think to take these kinds of measures.