Committing even a single evil act loses paladinhood. You can atone, but there is probably a limit to how many times clerics and your god will be willing to even offer the atonement (probably only get the quest one or two times at most). Committing evil acts tends to drain away good alignment faster than committing good acts drains away evil alignment. Neutral is what you should end up if you commit occasional evil acts but do a bunch of good to try and make up for it.
However killing specific targets isn't necessarily good or evil. A paladin can kill good or neutral targets under a number of conditions, such as if they are controlled by evil and a paladin doesn't have a way to break this control or otherwise disable the target without allowing a greater evil than killing them to occur. They can also kill in self-defense, such as in a case where a good target is somehow convinced a paladin is their enemy and won't give up, or neutral targets are attacking them such as by being members of an enemy nation to where the paladin comes from.
You could even have two paladins wind up killing each-other in a holy war of some sort, although the conditions for this to happen would be immensely complicated for it not to cause either of the paladins to fail their oaths.
Killing being a sin is a fabrication of modern Christianity and a few other religions, and not relevant to the Forgotten Realms or most other D&D settings. Murder might be a sin, but not all killing is murder.
Some acts will be seriously super evil, and should seriously impinge on a character's alignment. Others are just being a jerk, and should have a minor effect, if any. Paladins are allowed to be jerks. Good is not necessarily nice. Good doesn't mean stupid either. Good individuals will however place their life in danger for the good of the many. They are not obligated or expected to be able to save everyone, or to sacrifice themselves when innocent hostages are threatened, and stupid stuff like that, and they do have to consider the value of their own lives in furthering future good when taking on dangerous enemies. They do not, however, possess fear, they merely need to consider the future, and if they can really do more good by sacrificing themselves now than all the good they are needed to do and can do in the future. Being practical and not suicidally self-sacrificing shouldn't impinge on your alignment, depending on the circumstances. A level 1 paladin isn't expected to face an evil army or a great wyrm red dragon on their own. A 20th level paladin with a party of similar strength would however be expected to deal with the aforementioned army and/or dragon.
Things are a bit more flexible for other classes in terms of alignment than for paladins, so you need to track doing evil acts (and other acts against paladinhood, like lying) at all separately from your general alignment. A lawful good cleric won't get in trouble for lying a few times, unless it is a specific prohibition of their god, or it otherwise pisses off their god, at least not unless they lie so much and/or in such ways that their alignment changes.
Exalted characters are similar to Paladins in these ways if you implement the Book of Exalted Deeds stuff, they also lose their abilities immediately upon a single evil act, or when violating certain specific prohibitions to their various forms of exalted states, like Vow of Poverty, or Vow of Chastity.
I think a good way to handle this might be to have some acts get a secondary 'EVIL' or 'non-Paladin' marker, some acts, like killing good or neutral enemies that attack you, could still give you evil points, but won't necessarily affect things like paladinhood unless done consistently without making up for it with good acts or just doing a ton of it, other acts however are clearly evil not matter how you put it, and should cause massive evil points for any characters who do them and immediate violation of paladin oaths.
Also, being nice or being a jerk should not necessarily affect your alignment. Especially in the case of charismatic or deceptive evil characters, or some lawful evil types, or jerk with with a heart of gold good types, except in some more extreme cases of such.
Killing non-evil targets when not in self-defense or defense of another should probably count as an evil marker type act, killing in self defense or defense of another will be a lesser evil that only is evil points.
Good points should require some degree of service or self-sacrifice, not just being nice, but going out of your way to help others, although not necessarily without gain. You shouldn't demand it, that would be merely a neutral act, but asking for a reward for something you'd do anyway or already did shouldn't negate the goodness of it unless there is a threat involved, although it might mitigate somewhat the goodness of it. Paladins need to eat too, and need to equip themselves, and their parties aren't necessarily as pure as the Paladin themselves.
It should take extreme circumstances for a paladin to willingly adventure with an individual they know is evil, and doing so would violate their paladin oaths except under very complex mitigating circumstances, and only so long as the individual commits no evil acts while the paladin is adventuring with them. PC paladins should be allowed to refuse party members of Evil alignment.