Arkham Horror (Third Edition)

Yes — even if modifiers reduce your skill value to zero you still roll a minimum of one die to resolve the test (p. 9).

When defeated you remove your investigator token, discard your cards and tokens, and place one doom on the scenario sheet; then you choose a new investigator that hasn't been played yet and, after the mythos phase, place them at the scenario's starting space and continue (p. 9).

No — while you are engaged with one or more monsters you can only perform the focus, attack, and evade actions and you cannot resolve an encounter (pp. 11–12).

The attack action engages a monster in your space (place it by your investigator sheet if it isn't already) and then you resolve a strength test modified by the monster’s attack modifier; you may choose a monster in your space even if another investigator is engaged with it (p. 11).

No — each investigator draws two tokens one at a time and does not return them to the cup; when the cup is empty, return all mythos tokens being used this game to the cup to replenish it (p. 14).

For each success on an evade test you disengage and exhaust one monster, and if you roll enough successes to evade all monsters you were engaged with, you may perform one additional action that turn (p. 11).