It is well known that each divine has a sphere of interest - Julianos is God of Wisdom and Logic, Dibella is the Goddess of Beauty, etc.

I've read the entries on the divines in the Imperial Library and on the UESP, but I'd like to get some advice from lore experts regarding fuller details of their teachings and how their interaction with pilgrims works in lore. I will be putting this into practice as part of an overhaul of shrines in Skyrim. Each shrine will grant a blessing based on how the player has followed their teachings.

If a pilgrim was to make a donation to receive a blessing at a shrine to any of the divines, would there be any cases where the divine would refuse a blessing? For example, if the pilgrim has had dealings with daedra, would this affect how the divines look upon the pilgrim?

More specifically, how would each divine reward a pilgrim that has followed that divine's teachings?
For example:
Mara is the goddess of Love and Peace: If the pilgrim has assaulted or murdered people, I assume Mara would not look too kindly on the pilgrim. If the pilgrim has just killed during combat in self-defence, would this effect her stance?
Akatosh, dragon-god of time: If the pilgrim is dragonborn, and has killed many dragons, would this affect his stance? If the pilgrim has killed Paarthurnax would this affect it?

Why would people pray to a particular divine in the first place - if they are scared of displeasing the divine, how exactly would they displease that divine? What practical benefits would there be for a pilgrim by serving a particular divine?

Would a divine go so far as to curse a pilgrim who has strayed?

If we go by lore and not gameplay mechanics then we know that Aedra reward those that are faithful to them with blessings, visions, artifacts and the like.
Conceivably they would also refuse to bless those that stray too far from the dogmas of the temple.

Not a lot is known about these things, so for roleplaying purposes youre free to say youre character cant get Mara's blessing due to this or that.

People seem to mostly follow the cult of a Divine they were born into, just like people do in real life.
Conversely, it should also be possible for someone to adopt a new patron deity.

As far as I can tell pilgrimages in lore are mostly to show piety to the temple, not so much the Divine.
Pilgrimages seem to mostly retrace the steps of a past great hero or saint, symbolically re-enacting a victory or insight.

You follow the divine you feel most comfortable with, or the one you like most. There are a lot of things that could please a divine or would not please him/her. A list for as far as I can think of anything:

Akatosh:
Likes: Killing dragons, beating Alduin
Dislikes: Nothing

Arkay:
Likes: Solving murders, giving people rest
Dislikes: Black Soul gems, necromancy in general

Dibella:
Likes: Things to do with beauty, art, six
Dislikes: Things that would stop that (like http://www.uesp.net/wiki/Skyrim:Caught_Red_Handed)

Julianos
Likes: Justice, so doing lawful quests or cleaning up a city (destroying the DB, killing Thonar Silverblood)
Dislikes: Crimes, so Thieves Guild & Dark Brotherhood

Kynareth
Likes: Nature, Thu'um. So do nature things. Save animals and learn more words of power.
Dislikes: Crimes against nature. So don't kill any bunnies.

Mara:
Likes: Love, do her quests
Dislikes: Things that stop love (murder)

Stendarr:
Likes: Mercy, hunting Daedra
Dislikes: Worship of Daedra, not showing mercy

Talos:
Likes: Killing of enemies
Dislikes: Running away from them

Zenithar:
Likes: Creating, buying and selling items
Dislikes: Hoarding or bad deals. Probably also dislikes theft

I'd assume that if you meet certain requirements they would stop blessing you though. It would make sense.

Pardon, but why would Akatosh like someone killing his "Children"?

Cause he supposedly blessed the dragonborn, why would he do that if he did not want you to kill dragons?

Not all dragonborn is dragonslayer, to my knowledge Alessia, Reman and Tiber have never slay a dragon, on the opposite Tiber has a dragon ally Nafaalilargus.