Azlan said:
1. Does a cleric have to wield – as often as possible, in combat – his deity's favored weapon?
Nope. Not required. Just a minor, personal preference of the deity's clergy, it has no impact on what a cleric is allowed to wield.
3. If I choose to be a cleric, but I do not choose the War domain, do I still gain proficiency with my god's favored weapon even if that weapon is not among the cleric's weapon proficiencies?
Nope. If your deity's favored weapon is a Simple weapon, like the Quarterstaff or Heavy Mace, then you're proficient anyway just because clerics are all proficient in Simple weapons. If your deity's favored weapon is a Martial weapon, like the longsword, you
only gain proficiency in it if you take the War domain (and only if your deity has the War domain listed among his/her/its domains),
OR if you multiclass into something that does have Martial proficiency (like multiclassing into fighter) (QUICK EDIT: or if you take the appropriate Martial Weapon Proficiency feat, of course). If your deity's favored weapon is Exotic, then you only gain proficiency in it if you take the appropriate Exotic Weapon Proficiency feat (which requires at least a +1 Base Attack Bonus beforehand).
4. What are the consequences, if any, a cleric suffers if he does not prefer to use his deity's favored weapon?
Nothing. Pelor doesn't care if you wield an Orc Double Axe, or a Quarterstaff, or a Longbow, or your bare hands. Pelor's dogma does not say anything about denying you access to other weapons. Neither does any other deity's, to my knowledge.
Only particular advantage to using the deity's favored weapon is a) if it's from the War Domain then it's just the benefit of a free proficiency and Weapon Focus, and/or b) other followers of the same deity might have an inkling more respect for you if you wield the deity's favored weapon, just because it's a bit more traditional for that deity's clergy in emulation of their god/goddess.