Why should the law care about what I do behind closed doors?

It shouldn't to a point.

You can stand on a public sidewalk and take pictures of my house. You can't walk into my house uninvited and start taking pictures.

The law doesn't care about what you do behind closed doors as long as it is within the bounds of what the law allows.

Why do people keep saying they have evidence and have presented it that proves you're wrong even though they have none and haven't presented anything? Furthermore, what do they think you're wrong about?

Society sets laws announcing those actions that it deems unacceptable in polite society. If evidence appears that causes a reasonable person to suspect that illegal activity is going on, society should investigate. Of course society might find itself having to jump through hoops by adhering to constitutional law. It cannot just invade your personal space and demand to know what you're up to just because they don't like you.

If evidence arises that you are doing these things behind closed doors, don't you think the government has a moral obligation to investigate?

Liberty is not boundless. It does have its limits.

Why do trans people get so deeply offended when a stranger misgenders them, especially when it's a first encounter? I've been socially transitioned for 4 years and it just feels like a waste of energy to be so hurt by it.

But what if you're raping little girls behind closed doors? Killing gay men? Watching child porn?

Your speech is free. But if it causes malicious harm to someone, you can be sued.

The law shouldn't care (if you are a law abiding citizen) about who you take into the bedroom as long as they are consenting adults. How many guns you own. What you eat for supper. What kind of TV shows you watch. Whether you watch porn or not.

Do crossdressers like wearing pantyliners and tampons in their butts?