An investigator uses a typewriter to damage a monster (there are no "weapon" or "attack" keywords). Is it an attack (and thus does not require an Evasion)? If so, is this a combat check (and thus is penalized by Darkness)?

Its the same as using the Lantern in close combat. You need to roll an evasion roll against every monster in the same space.

It is clear to me that the Lantern and Typewriter is not intended as a weapon, since it clearly only say Equipment on the card. Using an action, an investigator can apparently use it as an improvised weapon, dealing damage to a monster. But no combat is ever initiated, so you don't take combat penalties for darkness and stun etc.

At least that's my opinion.

Page 23, "Combat Keywords" lists "Deal X Damage" as a potential keyword, hinting that "Action: Deal X damage, discard card" is a combat action. Imho this only by-passes the whole "draw combat cards" and randomization thing, so you don't end up with strange descriptions.
Further hint: You deal damage to a monster

We had a somehow related question ("do you have to evade monsters if you don't attack them but OTHER monsters in their space") here a couple of weeks ago.

Q: Does an investigator in a space with a monster have to evade that monster if he intends to attack a different monster in a different space, e.g. using a ranged weapon?

A: As per the rulebook, he does not need to evade to attack another monster (page 15):
Attacking a monster does not require making an evade test.This attack may be unarmed, with a weapon, or with an “attack” Spell card. This attack does not need to target a monster in the investigator’s space.

I hope this answers your question!
-Corey Konieczka

So Corey states that there are several different types of attack actions. Remember that every ranged attack is initiated by selecting the "Action:" part of a ranged weapon.

That said, there is no rule explaining how equipment-that-can-be-used-as-weapons work in combat right now. Personally, I deal with it as "if something damages someone, it's combat". But that's my opinion.

You might want to write a rules question (link is at the bottom of the screen) to FFG