having alliances with non catholic factions and relation with pope?

[Aron]

Does allying non catholic factions lower the relation with pope?

[Hyaweh]

unless you're at war w/ other catholic factions, or the pope hates that factions, prolly no

[Tears of Destiny]

Having trade right or any other kind of diplomatic relations with Faction X that Faction Y hates will lower it slightly.

Same with building opposing religious buildings.

[Aron]

Having trade right or any other kind of diplomatic relations with Faction X that Faction Y hates will lower it slightly.

and vice versa
Having trade right or any other kind of diplomatic relations with Faction X that Faction Y like will raise it slightly.

Thanx lads

[Meneth]

Yes, alliances with non-Christian factions will reduce standings with the Pope:

Trigger Alliance_Declared_Catholic_to_Non_Christian
    WhenToTest FactionAllianceDeclared

    Condition FactionReligion catholic
    and not TargetFactionReligion catholic
    and not TargetFactionReligion orthodox

    FactionStanding factions { papal_states } -0.1
    FactionStanding target_faction normalise 1.0 100
    FactionStanding target_allies normalise 1.0 200
    FactionStanding target_enemies normalise -1.0 200

However, only making the alliance reduces the relations. Ongoing alliances do nothing directly.
On-going alliances makes enemies of your allies and allies of your enemies dislike you, though, so if the Pope is at war with one or more of your allies, relations will deteriorate.

"Having trade right or any other kind of diplomatic relations with Faction X that Faction Y hates will lower it slightly."
This is incorrect, only alliances affect diplomatic relations over time.
Trade agreements only affect standings with who you're trading with, and their allies, and only when the trade agreement gets signed:

Trigger 0079_F_Trade_Agreement
    WhenToTest FactionTradeAgreementMade

    FactionStanding target_faction 0.1
    FactionStanding target_allies normalise 0.6 50

[Aron]

Thanks, appreciated.