I just started a new game, and I can't progress through the Dragon Rising quest. Two problems pop up:

1. Irileth leaves Whiterun, goes to the meeting spot near the Western Watchtower, then goes back into the city.

2. Nothing happens after killing the dragon. I loot the corpse and stand there waiting to absorb its soul, but nothing happens. I've tried fast-traveling to another area and returning to the Watchtower, and that didn't help.

I use the unofficial patches for the original game and the three DLCs. I thought those were supposed to fix problems like this.

Sounds like you're not the Dragonborn. There's plenty of room in the world for people who aren't Dragonborn, but you may want to hit the tavern for a few drinks to drown the disappointment of finding out the hard way.

If there's a way for you to reload to a save before the quest even begins, you may want to take that path. If not, then just loading to before fighting the dragon may work. This is a glitch that happens sometimes with dragons regardless, but it's unfortunate if it happened randomly to you. Is this a recurring problem, or did you only try this once?

Can you post your mod load order please.

Not being able to absorb dragon souls is a known issue. Last I heard the fix is to have the Unofficial Dragonborn Patch. You should have all the unofficial dlc patches.