Well, I'll bite. As fair warning, this is another outsider perspective, as I'm from England (Land of 'It's damn near impossible to get legal permission to own a firearm out of a few very VERY specific circumstances' and as far as I'm aware the trans bathroom deal is not an issue here, people use the bathroom of their gender (which is a matter of identity.) )
I'll go over my stance on the issues presented but then I'll try to speak about centre/left/right politics as a whole, because I do believe that's how this topic was intended.
So on the trans bathroom thing, I very much stand where my country (and it appears pretty much everyone in this topic) does. People should use the bathroom of whichever gender they identify as (if applicable). Oftentimes here in England, much like a lot of the rest of the world, politics devolves into 'Which side can make everyone more scared of the other'?. It is unfortunate, but fear is a big part of how politics works. Frankly, it sounds like an entirely reasonable fear for people on the other end to have to, and I'm quoting someone I've had this discussion with irl: 'Let people with man bits in lady bathrooms and people with lady bits in man bathrooms.' (You can just tell that was a FUN discussion.)
However, while that does seem like a reasonable fear to have, I do also consider it unfounded and often born of ignorance. As I believe has been said above, it is frankly not all THAT difficult for a man to get in convincing enough drag that he won't be questioned going into a woman's bathroom, or vice-versa. If some scum wants prey on people of the other gender in bathrooms they already have tools to do so. Even if this would add another such 'tool' I doubt it would really change anything. It seems very strange to me that people would want to inconvenience and discriminate against an entire portion of the population for such a reason (And I don't recall ever hearing a case of someone claiming they were trans to get into the other bathroom and then assaulting someone over hear, although I can't say for sure whether or not it has happened. But I feel like if it was a huge problem warranting seriously considering telling trans people they have to use a particular bathroom, an average person like me would have heard of at least one case of it happening.)
Alright, on to gun control where I think my stance is a little more nuanced. Now, as stated above, we have very strict gun control laws over here. You very rarely hear about any level of gun violence here, and when you do it's almost always tied to organised crime (And as pessimistic as it is, I doubt any level of gun control will prevent organised crime getting the weapons they need). As far as I'm aware, as a country we've only ever had one school shooting (after which we pretty much banned handguns). Frankly, I would never even CONSIDER supporting wider access to guns in England. But the discussion is not about England, it's about America, and that's where the greater nuance comes in. A while ago, people over here were having big discussions about the possibility of Nuclear Disarmament. While I won't go into my particular views on the issue, one very reasonable viewpoint that often came up was, simply 'Until nobody has nukes, everybody needs them.' That attitude still has credibility in the gun debate.
The fact that guns are so widespread already is an issue. Plenty of people have them and it's oft a part of your culture. I am not aware of exactly how easy it is to get a gun in America, and I think there should absolutely be strong checks in place with this sort of thing, but cultural differences are a big thing. Sure, to me, extremely limited or no guns seems the obvious way to go, but that's because that's how things are and pretty much always have been here during my life. But I'm well aware that big changes to things which are potentially ingrained in society or culture are difficult to get right, no matter how obvious of a change it may seem to an outsider like me. Put simply, I don't really feel I would be able to offer a good view on this without a better understanding of American culture.
I think the question of politics itself is a really very interesting one. I've said where I stand on these issues which is largely aligned with where my country does, because that's what seems reasonable to me. However saying where I stand in regards to the centre/right/left is trickier. I tend to consider myself reasonably centrist, although I'm probably at the rightmost end of the centre. In the few opportunities I've had to vote in my time I've gone both ways before, depending on what was being voted on and what the candidates both have on the table, and just generally who I thought would do the job best. I would get into more detail there but I don't fancy explaining the British political system and the various things we generally vote on.
Now, the difference is, centre-right is not at all where I feel I would stand with American politics. I don't claim to get all of it and know everything that's going on, but as an outsider, centre-right in the UK seems to me to map to a pretty strong left over in the US. I recall doing a rather extensive quiz that someone on the forums (Jericho I think) posted which asked a hell of a lot of questions and matched you to which US candidates you most agreed with. I took it and found myself with very high (and in fact equal) percentage matchings with both Bernie and Hillary and very low ones for nearly everyone else. And with that perspective I find it really difficult to comment on the dynamics of these things because it looks to me like my right is a very different thing from your right, and the same with my left.
What I will say, however, is that while I do generally try to advocate what I see as the middle ground, I believe some situations do call for a potentially strong swing to the right or left. But doing so in other situations can be not only ineffective, but also sometimes dangerous.
I guess what I'm trying to say in my own really very verbose way is that I evaluate most political issues on a case by case basis, and sometimes I'm left, sometimes I'm right, and sometimes I'm down the middle, just wherever seems the most logical to me. Here's hoping I've not missed the point of the topic.