shifts the ward is worth, including triggers and duration.
Just for another PoV, in our game the shifts spent on duration, triggers, and any special conditions (wardflames, people that can pass freely, etc.) are not included in the difficulty to bring down a Ward by whatever method - just the final Strength.
(By not quite perfect analogy) bringing down the castle wall is a matter of the strength of the wall's stonework... arrow loops, battlements, murder holes, interior rooms and their contents, or light fixtures don't matter to the siege engine's payload as it comes crashing in.
Now, we could have easily gone with the other way (higher target number). We just chose the smaller number value to keep it more interesting. It allows NPCs that aren't on the 13-14+ refresh level to actually still be a threat and the Wards to be treated as just a single step of a layered magical security system, not the "be all, end all."
And to the OP's original question
Could you counter-spell them?
In our game we allow this, and taggable Aspects include assistance from other spellcasters doing applicable and generally Lore-based "magic-y" things that sound like they might boost the Counterspell (up to and including doing a weaker "counterspell" that counts as a Maneuver to hand it off to the main spellslinger making the roll).