The airline gets $351, $1 base fare plus their own internally decided surcharge of $350, here labelled "International/Domestic surcharge". The rest goes to other agencies. $19.72 for the security search at FRA is just LH's computers working the way they do, that may or may not be valid (it probably isn't here), they could/should handle that differently. Let's say that's an extra $10 on average. The last two charges should be labelled "airport user fee", so LH doesn't get them but LH does have a negotiation piece with the airports which could enable it to go higher or lower.

Either way this is something like $360 fare, $176 taxes and fees. The $350 is not avoidable by booking at a counter, it's just part of the base fare but expressed differently.