Ok, the Chalice appears after a stage timer has been close to running out twice. The 1UP is still eluding me. If I set a certain enemy/object state to $16 (which also freezes it) and get hit by it, it'll increase the counter. Not sure what actually sets that state organically in the game - I'd have to play through it on emulator. It's not the clock :)

Seems that all the tree/broccoli does it ensure you get 4 food drops after clearing the stage

A bit late to the party here, but I've been looking into the code of the game, and this is the only place I've found on the internet that actually discusses these things. :P

Of course, a lot of stuff already covered here just makes my findings irrelevant. :P Though a few things I know that haven't been mentioned.

The full item priority order is:

Hearts > Chalice > Door > Pencil > Crystal > Star rod > Shoe > Candy > 1UP > Clock > Cane > Tree > Umbrella

If one of those items is queued up, it will always appear instead of the next in line. That's what makes umbrellas a lot more rare than they should be - crystals get queued up every 68 seconds, so they usually steal the priority.

The extra credit PENCIL appears after clearing 20 stages, so it's pretty unavoidable outside of skipping stages. The DOOR appears after grabbing two crystals, so if you're trying to let them time out, it shouldn't appear. But of course if you're trying to time the appearance of the star rod it could be an issue. The TREE appears after taking 20 hits (whether you had extra hearts or not). The CLOCK appears after picking up 7 (or 8? first one sets the counter to 0) of the same type of food item (like the hearts, food dropping from the sky don't count), and picking up another item type removes the clock from the queue. This is why one of the only places you'll usually see it is on Round 11. The CANE is triggered by bursting four or more enemies simultaneously, but the counter is constantly reset to 0, so it requires very precise timing and no other items spawned on the current stage yet, which makes it extremely rare. Probably much more rare than intended. I've only seen it on the first stage because no other items were queued up.

The game works a bit different from other games in the series which decide the item to spawn when a stage starts. This game checks periodically if an item is queued. So if no items have spawned yet and you jump enough times on a bubble, a shoe will appear on the same stage.

Still unsure what triggers the chalice or the 1UP, but threshold for both counters is 2.

Also not sure what the tree does. Anyone know?