https://pathofexile.gamepedia.com/Elemental_damage
Elemental Damage refers to fire damage, lightning damage and cold damage collectively.
image of my character screen (C):
Note that the fire damage isn't displayed either, while cold and lightning are — does my character lack this specific type of damage entirely, despite using same fire spells?
One item I'd like to equip seems to lower the elemental damage (the values are displayed in red), and I don't know how strong the effect will be.
There is no separate line for elemental damage in your character sheet.
You are correct that the skill shown in your character sheet deals no fire damage. Note that your character sheet has all your active skills listed. Double check that you are looking at stats for your primary damage skill, not the default attack.
Your new weapon likely deals fire damage. Fire damage numbers will be displayed in red (colour code for fire), just as cold damage will be blueish and lightning will be yellow. It does not mean your fire damage will be lower.
