The phrase "get to do something" implies to have the opportunity to do something. So the meaning of the first sentence is "I love to have the opportunity to see her first thing in the morning".
The phrases "get to do something and get to doing something" aren't interchangeable to convey the same meaning. The latter phrase means "to start doing something such as I got to talking to her. I got to thinking that it was his fault.
Both the sentences are grammatically correct, but they convey different senses.
Yes, they have different meanings. In the context of Fifty Shades, I don't do romance means the character is wholly uninterested in the usual trappings of attracting a mate: wining and dining, emotional intimacy, evoking tender emotions, etc. The speaker is solely interested in sexual activity. On the other hand, someone who says I'm not romantic might be interested those things but not very skilled at them.
I don't do X means I'm unwilling to engage in X. It connotes unwillingness and does not generally convey inability. See the extremely topical question "What does 'I don't do dogs' mean?" for an in-depth exploration of this phrase. This is a modern construction, so it's unsurprising that you didn't find any hits in your N-gram search.
I'm not romantic expresses a more fundamental statement about the person's nature. Romance is alien to the speaker's personality and emotions. It would be difficult, and perhaps impossible, for them to experience or express it.
There's also a subtle difference between romance and romantic. In this case, romance is an activity, while romantic is a descriptive quality. An unromantic person might well be involved in some romancing, but it would likely either be awkward for them to attempt (because romance is not part of their nature), or one sided (someone else attempts to romance them, but encounters difficulty, because they don't experience romantic emotions much or at all).
Conversely, someone who doesn't do romance may be fully capable of experiencing romantic feelings, but they would surely stifle and not express them. Likewise, they would categorically reject all attempts to romance them, whereas someone simply unromantic might warm to the idea, or at least appreciate it.
Best Answer
The difference between them is that nevertheless is more appropriate in reference to events or things that might 'never' happen, whereas nonetheless is more appropriate in reference to measurable, quantifiable things that you could say there is 'none' of.
For example:
'Leaving' is an event that could or could not (never) happen.
This comments on how much he enjoyed something, a measurable amount.
I feel that both your examples require 'nonetheless'. The first is asking whether something is more or less misleading than something else, so it is about measurement. Likewise, the second example compares two kinds of welcome, and says one was no less of a welcome than the other.