First of all, you seem to have a lot of anxiety about how do you come across. Been there, done that. You may also have some of the mild psychological disorder, but it seems all too common to throw around diagnoses without looking into other possible causes of confusion about Shit People Do.
'Be yourself' annoys the hell out of me, it's tired and cliche, but there IS a point in that. The point being, fuck the world, if you like it and nobody else may get hurt, do it.
This is harder than it sounds, really, there is a sort of an instinct to fall back in line, but if you want to get anywhere, remember that people will think of you whatever they will want to, regardless of reality. Humans aren't really optimized for being rational and just in their judgments.
You say you're a nice person, I say you're an unassertive person. 'Nice' is an interesting word. It is basically a piece of linguistic candy, an equivalent of saying 'Good dog'. This kind of 'niceness' is just a display of subservience. (low status behavior too, so if you try to be a people-pleaser don't expect respect). You are acting like that either due to ingrained belief that you should act 'nice', or the need to validate yourself as a human.
If the former, if you want to be nice, act out of respect, not compulsion. If you value your friends, giving them a gift or doing them a favor should come naturally, without it feeling forced on you. If you'd feel forced to do something for a friend, say no in as polite manner as you can, and escalate should you be forced more overtly. Worth mentioning, someone who is trying to force you to do something is likely not to actually be a friend.
If the latter, it's the 'ugh, the issues' part. Remember that you are staying alive out of your own and no-one else's will (how you came to be, of course, was not up to you to decide), so you are ultimately the only person who makes decisions about it, and that any opinion someone make about you is just their judgment of you, not a cosmic truth.
Also, most of the time it's not being nice that matters, but not being an asshole. People actually don't give a shit about someone being nice, they get used to that and the result is that you are treated as a servant in all but name. So, long story short, stop trying to act nice, do not stop not being a jerk.
Now, moving on to other points: professionalism is a really nifty facade. It's mostly an image thing, really, and beside that part, it's not about focusing so hard you get tunnel vision, but treating things seriously. Like you said, 'game face'. That's just a bunch of applied tricks that basically convey 'Unless you're involved in my work (as in, the actual things you do, not the job itself), fuck off, I care about what I'm trying to do'. Quite useful, but there's a time and place for everything.
People talk about politics and religion and so on not because they want to piss each other off. Theoretically, they are trying to convince the other people that they are right, and get them to do things they think are right. Actually, this may be true, but just as often it's just a display of allegiance to a 'tribe', being people sharing their religious, political, etc. views. You don't have the weaponry to deal with those, because you haven't dealt with finding the answers to the underlying questions, and once again, you fear upsetting others, so you are unwilling to defend your position.
About the tone, I fear I cannot help, since inability to perceive tone is just as alien to me as, say, being deaf. I know some people are, I just can't comprehend how does it feel.
Also, @Imperfect, I've grown to dislike the Introvert/Extrovert Jungian categories, since it makes you entrench yourself firmly on one end and point machine guns at opportunities. Either it is wrong, or by some miracle I landed on the exact middle between the two, since I can express either Introverted or Extroverted traits and behaviors, and my energy is a bell curve over time in both social setting and while alone (i.e. after a while my Alone alarm turns on and I have to talk to some people, and after a while my Social alarm turns on and I get a reeeealy nasty attitude towards the world in general).