How Rushdown, Staring Contest, Perfection and You Are Winner unlock

These four unlock from the fight clock, the tries left after round 18, and the trophies you have won. A History refight, a lobby and a run with custom rules do not unlock them.
Rushdown
Rushdown unlocks on a win while the fight clock is still under 3 seconds. A win at 3.0 does not count. The clock starts when items activate, and the number on that clock is the one that matters.

Spear deals 3 to 8, costs 1 stamina and waits 1.5 seconds. Its first swing can land inside the window. A second swing at 3.0 cannot, because 3.0 is not under 3. Two such hits still do not, by the listed damage, clear a full opening health bar. Bismuth Cube watches one affected item. Every 7 buffs you gain, it moves that item's cooldown forward by 0.2 seconds and grants a random buff, up to 10 times. That is a small advance. It does not by itself win a fight in under 3 seconds.
Staring Contest
Staring Contest unlocks on a win once the fight clock is at 45 seconds or later. A win at 44.9 does not count. A win at 45.0 does.

Fatigue deals its first damage at 17 seconds and then every second, to both sides, and the amount grows. Healing Herbs and Banana are healing you can still be holding. Banana heals 4 on a 5 second cooldown. Wooden Buckler rolls on a melee hit and can subtract 7. Lasting until the clock shows 45 is the condition. It is not a prediction that this trio wins.
Cooldowns, buff ticks and fatigue is the timing of that clock.
Perfection
Perfection unlocks when you finish round 18 with 5 tries still left. Those hearts are tries, not combat health.

A run starts with 5 tries. A loss spends one. One can come back when the round counter reaches 8, and one can come back when you choose to play on after 10 wins, each time only if a try remains and you have fewer than 5. Finishing at 5 is the test. Stop at 10 wins or play on to round 18? is the choice those tries belong to.
You Are Winner
You Are Winner unlocks when the trophies you have won go past 10000. Exactly 10000 does not unlock it. Spending trophies lowers the balance you can spend. It does not lower this count. A History refight does not add to it.

Each of these four pays 10 trophies when its popup appears, and that payment counts toward trophies won when the unlock is allowed.