Does the 3/8 attack effect created by Lord Jaraxxus count as a weapon

No there is no way to counter armor with cards at the moment.

To counter-play against an armor stacking opponent:
- Be aggressive
- Do not stall the game

Here is a quote:

Armor is a form of secondary Health available only to heroes, gained through certain spells, hero powers and minions. Damage taken will always be deducted from Armor before Health. Only when a character has zero Armor will they lose Health from a damaging attack. Armor can effectively extend a hero's Health far beyond the 30 points otherwise allowed.

The following is for informational purposes only, as it only applies to Hearthstone prior to the patch on 11 March 2014 which bug fixed how armor and minions work.

Here are some extra useful niche information:

While Armor serves as a substitute for Health, for game purposes losing Armor does not count as suffering damage. For example, losing Armor (but not Health) to a Water Elemental's attack will not cause the hero to become Frozen, and loss of Armor will not reduce the summoning cost of Molten Giant.

Source: http://hearthstone.gamepedia.com/Armor#Armor

The following is for informational purposes only, as it only applies to Hearthstone prior to the patch on 10 Dec 2013 which bugfixed Alexstrasza.

The "15 health" used to refer to the "current health + armor" total.
Scenario 1: Target Warrior has 28 health and 13 armor. When Alexstrasza targets this Warrior, Alexstrasza visibly deals 26 damage (you see the -26 number) to the target Warrior to result in 15 health and 0 armor left.
Scenario 2: Target Druid has 9 health and 3 armor. When used on the Druid, he will visibly be healed 3 health to result in 12 health and 3 armor.

Yes, if a minion attacks a hero with the weapon exposed, they will take the damage from the weapon, but the weapon will not lose any uses.