Yes, being a Paladin is almost impossible in this game because there are some quests where you literally have to do something evil and trap you until you do it.
I think Bethesda really lost the plot here with optional choices. Previous rpgs including Bioware`s baldur`s gate, NWN, etc would have an alternative, somehow, somewhere even if it took longer ad was harder.
I dunno what`s going on with Skyrim.
In this game you can only play only a Neutral character who tries to do good.
I think it's still possible to be Lawful Good in Skyrim, the tough part is in accepting that things aren't going to go the way you want them to most of the time. It's a dark world built towards morally grey characters, and in playing a strictly good-aligned character like a Paladin you're going to see some dilemmas.
Point is, you must constantly try to obey the laws and promote good whenever possible, try to make the best out of all situations. In the end, Lawful Good is an idealistic alignment, you do the best you can for the world, even if things don't turn out well (especially so in a gritty world like Skyrim).
The true problems arise when you are forced to make choices and aren't given enough options when there should logically be others, which is Bethesda's fault. Those situations are unfortunate and jarring, and you may have to RP praying to your patron for atonement, but thankfully they aren't overwhelmingly numerous.