What is the easiest way tweet a random post from a category of my website each day? I really don't want to use RSS, is there another easy option?
RSS Feed is your best option
I am not super familiar with this method. For RSS in general across all-platforms: Will it only post the latest RSS feeds?
It will extract latest X posts, of course you set the X in RSS options in Mass Planner. Then it will extract X new posts from that article every 240 minutes (that's default value but you can change it Overview tab > Advanced settings). All posts will be sent on Posts list. You can control how many posts per day you want to publish in When to publish tab of the campaign.
Lots of feeds don't display every post. This site only shows the 10 most recent posts..
That's how feed works. They'll show you X latest posts. Wordpress websites have 5 by default. If you own the website, you can change that.
It's up to webmaster, nothing you can do.
Create CSV with all posts and add them that way. Use randomize on the posting screen.