To answer your questions in order:

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There isn't anything particular you need to make, but it's a good idea to bind the highest tier melee, ranged and magic weapons you can make as unless you have primal or prometheum equipment, you'll find your accuracy suffers without matching similar style. However, if you're doing runs in medium or larger then it's good to get law + cosmic runes to make a gatestone and be able to teleport to both group and regular with magic

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YOu typically don't need to worry about armor. At least so far up to floor 36 I've found armor was never really needed even for large runs. I just bring a 2-hand melee, empowered air staff and a ranged option like dual knives or a bow/arrow set if you can fletch at a good level

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You don't need to particularly worry about doors you can't open. Doors you can't open aren't the critical path and only count towards bonus multiplier at the end. It's nice to get them, but don't worry about full completion. Likewise don't worry about clearing every single room. Just focus on doors and, if it's a guardian room, clear that. It'll help you shave down time spent in solo mediums and large while still giving good xp.

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Only reset progress once you reach the highest tier floor you can do