Making this to post general strategies and glitches that would otherwise be lost in Discord.
Here's how to do grate skip:
Start walking towards the screen where the grate is. Pull out the crowbar before the screen transition and hold Enter. (It's important that you hold Enter before the screen transition and keep holding it through the transition as well.) Hold Enter over the grate so you get the input on the first frame possible. Let go of Enter, then right click to get to the hand, left click.
As best I can tell, here's what happens in the castle RNG:
When the room loads, there's a random chance for the blue beast to appear. In the organ room, this is 50%, and in the north / south dining rooms, it's 75%. If you fail this check, the meanie will never show up, until you leave and come back, rolling the dice again. If the check succeeds, then the following things happen:
If you're in the north dining room, wait 1 cycle (1 frame of animation, as I understand it). If you're anywhere else, wait 2 seconds.
If you're in the organ room, wait randomly between 2-4 seconds. If you're anywhere else, wait between 4-10 seconds.
At the end of the wait check to see if Graham is above or below certain x / y coords. For example, in the picture below, you need to be above the red line.
If Graham is there, the blue beast appears. If Graham isn't there, start over at step 2.
Mordack's castle RNG continued:
In the organ room, you need to be between these two red lines at step 4:
And you have to be to the right of this red line in the south dining room:
Here's the science on the cat showing up in the bedroom. When you enter Mordack's bedroom, the following steps happen:
Step 1: Generate a random number between 0 and 100. If it's over 90, Mordack shows up to choke you out. Otherwise, move to step 2. Step 2: Generate a random number between 0 and 100. If it's over 20, call procedure 550_16. Otherwise, do nothing. Procedure 550_16: If the cat isn't already in the bag, generate a random number between 0 and 100. If it's over 20, make the cat show up and position it according to what room you're in.
So there's a 9.9% chance of Mordack showing up and a 55.9% chance of the cat showing up.
Oddly, procedure 550_16 has position coordinates for rooms where the cat never shows up: the kitchen, the organ room, the library. If those rooms actually had code that called proc 550_16, then it would be possible to encounter the cat there.