My Orc battle-mage likes to use summoned weapons and summoned atronachs to fight. Is this a no-no for a follower of the divines? Arkay in particular? I am guessing Arkay doesn't like necromancy at all, but what of the bound weapons?

In Morrowind conjuration was actually one of the imperial cult's favored skills

Bound Weapons and Armor should be fine. Indeed, an interesting crusader class could be made based around that.

As for Summon Creatures....the Familiar and Ancestral Ghost equivalents should be okay. Just avoid the Daedra and Undead.

IIRC Bound Weapons are Daedra summoned in the shape of weapons.

I'd say the Vigil of Stendarr would despise all conjuration magic, which could be traced back to the time of Alessian Order who seemed to suppress even all magic.

Beyond that, different divines may differ much. In fact I see no problem for a follower of Julianos to tap a little into Oblivion, as long as he stays with Julianos and his idea.

Hmm... Alright, but what of divines like Akatosh, and Arkay?

Aside from chomping on Dagons neck during the Oblivion crisis, I don't think there has been much to say whether or not his followers care about anyone using Antronach's and the like. Anyone following Arkay would obviously be against anything that tampers with the laws of life and death, but most Daedra and conjured weapons typically don't screw around with em. I don't think it'd be a big issue.

Arkay may more focus on anti-Necromancy. Though I don't think going to Daedra Prince not a valid path beyond the cycle, I suspect most Men would go against it. The dislike is more a cultural thing. None the less plain Daedra summoning itself is not violating Arkay's rules. Something you could use, like enchantments. But mundane abilities are above them, as we see Divines more fighter type (or, thief?)than mage except for Julianos. Akatosh may emphrasize mundane power more as the Plane(t) Akatosh is the Eye of Warrior.

Dont forget the little stunt his son did (Alduin) to him before he even became Dagon...

Would the divines be totally against using daedric armor and weapons in general? Non-summoned ones I mean. On another note, what of daedric artifacts like dawnbreaker? That weapon is used specifically to combat the undead. Seems like a righteous weapon in general.

In Oblivion, the Chapel Undercrofts were guarded by sanctified spirits.

I doubt that the Divines or their worshipers would be too fussy about generic Daedric gear. Certain artifacts, perhaps, but that the Emperor himself was said to have wielded one of Boethiah's artifacts, so unless it's something like the Skull of Corruption or the Mace of Molag Bal, I doubt anyone would care.

Supposedly priests of arkay are trained in conjuration magic to learn how to deal with necromancers more or chasing away wayward spirits and evil spirits.

As for Bound Weapons, they look more like magicka given form, rather than daedric spirits.

There's a big difference between conjuration and daedric magic. Daedric magic requires signing pacts with the daedra to gain power.
Conjuration is the art of overpowering a spirits will and forcing it to serve you. (Command spells were part of that school in Morrowind)
As a real world comparison you can look at medieval christian mysticism. Binding a demon to serve Gods work was considered white magic.