When I look at these artworks:

http://www.elderscrolls.com/skyrim/media/conceptart/word-wall/

http://www.elderscrolls.com/skyrim/media/conceptart/nordic-temple-ruins/

I have a question if Hagravens and Dragons were somehow connected long time ago. Someone said that they are creation of Hircine also that they could be his priests. But when I see those temple ruins in Reach with all the motives of birds and the original design of Hagraven - is it connected somehow or is it coincidence?

In The Legend of Red Eagle http://elderscrolls.wikia.com/wiki/Red_Eagle the Hagraven there is named as the Ancient one and Read Eagle/Faolan can spawn as dragon priest, also the name itself is again connected to that motive of temple ruins.

So it seems to me that those ruins are more like Forsworn ruins then Nordic and also that Hagraven could be connected somehow originally to dragon priests more then to Hircine?

OK, I found this:

http://www.uesp.net/wiki/Skyrim:Children_of_the_Sky

So it was really nordic ruins.

But still I don't really know who are Hagravens. The Red Eagle book mentions also Molag Bal, so I don't know.

I suppose one can never say never about any possibilities in Tamrielic history, but I doubt there is a connection between Hagravens and dragons - hagravens typically being associated with Hircine and dragons with Akatosh.

You're right to notice the avian influence on old Nordic architecture and there are a number of possible reasons for this. The bird influence could come from the Ancient Nordic pantheon of totemic animal spirits which contains two birds; the hawk and the owl. It could also come from a reverence of Kyne, later Kynareth, who being the goddess of storms (and the skies generally later), is often associated with or represented through bird-like/avian imagery. Stain-glass windows in Oblivion dedicated to Kynareth depict a bird and shrines to Kynareth in TES:V a decorated with an abstract bird-like ornament.

On the origin of Hagravens, deformed and grizzly fusions of crone and bird, I think it is very likely that they are a creation of the 'Father of Manbeasts', Hircine. Two organisations that feature numerous hagravens, the Forsworn (following the old ways of the Reachmen) and the Glenmoril Wyrd, are known to both have strong ties to Hircine.

Clan matriarchs and witches of exceptional proficiency in magic and an insatiable desire for more power probably pray to Hircine and beseech him to bless them with a boon. Hircine likely offers the magical power they desire if they pledge to give up their mortal forms and become (wo)manbeasts bound to his sphere. Like werewolves, hagravens are probably fated to go to the Hunting Grounds when they die.

As for some hagravens being associated with other Daedric Princes, I'd say it is not impossible. After all, a vampire (a creature originating from Molag Bal) could choose to worship another Prince or even the Divines if they were so inclined, but this probably wouldn't stop them from being bound to the King of [censored]'s sphere. Many hagravens are Reachwomen and, as such, probably worship a number of 'Old Gods' beside Hircine like Namira, Malacath or Arkay. But, Hircine would still be the god they are bound to by contract and their fate would be in his hands.