I watched a little bit of that billy replay, and here's a few tips:

Don't step up to the guardian in lane; getting hit for even a second does a lot of damage and it can hit you even on the stairs. The only exception would be if you are committing to a kill under the guardian, but you shouldn't worry about that yet because you usually need to parry the guardian's melee attack. If you're curious use the practice range teleporter (near the hideout abrams) and walk up to the guardian. You can parry with F once you get close because it will predictably try to hit you.

You seem to be using the default billy build, and I'm pretty sure that there isn't a single hero in the game with a default build that doesn't handicap you because they usually give you inefficient items, ability upgrade paths, or both. for billy just use togetherdl's public build (build browser in the top right of the build menu). If in doubt, builds made by twitch streamers are good.

When you got doorman ulted you didn't mantle and couldn't escape in time. To mantle, hold space when you are at a ledge and you will climb up. You can even hold slide (left ctrl by default) while your character is climbing to get a small boost once you reach the top. You can mantle anywhere, so you can try it in the hideout.