I uploaded two runs and got them verified, but they never appeared under latest runs. Are there any specific requirements?

They may not appear if they are old runs.

Its sorted of the upload date and games you follow are in there more often I think Either way, most users dont pay attention to it anyway, by streaming on Twitch to appear on the Streams page is generally better to get viewers

Your run was 5 days ago (actual run date, rather than submission or verification date), the oldest run I currently see on my front page was done earlier today. If you want a run to show up on the front page you pretty much have to submit it as soon as possible after doing it and hope it gets verified quickly.

That said, like @blueYOSHI said, showing up on the front page doesn't really matter at all. Playing the game live will get you a lot more attention than simply having a run listed beside dozens of other miscellaneous runs in a place that not many people look at anyway.

To be more specific, the front page show the most recent submitted run that were approved. So like shiken said, you need to hope it get approved as soon as you submit it which is rare except if theres a lots mod or if you are active at the same time as the mod

Also to specify what blue yoshi said, you can change your setting to see more or less thing about the game you follow at the top of the front page if you just go here

https://www.speedrun.com/Type your username here/settings/frontpage