Shinobi; while it's true that it may seem like blindfire is more accurate, it'll be less damaging.
"A defender using the barrier as cover receives a defense bonus for cover. If the defender is completely hidden behind the barrier, the attacker suffers a –6 Blind Fire dice pool modifier for not being able to see his intended target, but the hidden defender is considered unaware of the attack."
This supports my above reading; if you can see the attacker you get a defense roll, if you can't you don't.
However, you still have to shoot through the barrier if the defender is unaware. Read on:
"If the barrier takes the hit first, the gamemaster rolls Structure + Armor to resist the damage, and the structure takes any unresisted damage. If the Structure rating is exceeded by the damage it suffers, any remaining damage is transferred to the target behind the barrier."
So, not only does the structure roll Structure + Armor ratings (a minimum of 3 for glass, to the more standard 10 for furniture (hide behind the couch!) or even 26 for brick and plastcrete, but only unresisted damage exceeding the structure rating passes through.
Furthermore, penetrating weapons like firearms are affected by the rules on page 198, where a single bullet penetrating a barrier reduces the DV by one.
So, for an Ares Predator V using regular rounds with an assumed 5 net hits (even with a -6 modifier, it was a good shot!):
Base DV of 8P, modified DV of 13P, -1 AP
Against glass (SR1, AR2-1):
Roll 2d6>5, average 0 hits. Structure takes 1P (its SR), 13-0-1=12P exceeds its armor (AR1), rules for penetrating weapons for a single bullet reduces DV by 1, and 11P passes through.
Against drywall (SR2, AR4-1):
Roll 5d6>5, average 1 hits. Structure takes 2P (its SR), 13-1-2=10P exceeds its armor (AR3), rules for penetrating weapons for a single bullet reduces DV by 1, and 9P passes through.
Against ballistic glass (SR4, AR6-1):
Roll 9d6>5, average 2 hits. Structure takes 4P (its SR), 13-2-4=7P exceeds its armor (AR5), rules for penetrating weapons for a single bullet reduces DV by 1, and 6P passes through.
Against hardwood (SR6, AR8-1):
Roll 13d6>5, average 3 hits. Structure takes 6P (its SR), 13-3-6=4P does not exceed its armor (AR7), and the bullet is stopped clean.
As you can see, shooting through barriers is generally not a good idea, particularly if the target hides behind hard walls like brick.
Cyber-Dave:
Surprise generally only comes into play when combat is started; if, as in my example above, a running gunfight ends up in a PC/NPC hiding in another room (or otherwise taking themselves completely out of sight), then no surprise is generated. The cop would have seen the PC run into the room, and simply fires.
In my opinion, the PC escaping the cop to hide in another room should only have to roll surprise if he completely didn't expect the cop to shoot at him, or if it hadn't been a running gun battle already. This is up to GMs to decide for themselves, however.
See above for why this is generally a very bad idea, however.
To put things into context, even a heavy sniper rifle will have a hell of a time penetrating harder materials. Let's examine a Ranger Arms SM-5, which has 14P base damage, and an AP of -5, firing APDS rounds for an additional AP of -4, again assuming 5 hits for a modified DV of 19 (this guy is good!). It would still only take a brick barrier to stop the shot on average.
Brick wall: SR10, AR16-9
Roll 17d6>5, average 4 hits. Structure takes 10P (its SR), 19-4-10=5P does not exceed its armor (AR7), and the bullet is stopped clean.
Even a Panther XXL (DV17P, modified DV22P, AP-6) would be hard pressed to penetrate harder materials to damage the target behind the wall.
The same brick wall: SR10, AR16-6
Roll 20d6>5, average 5 hits. Structure takes 10P (its SR), 22-5-10=7P does not exceed its armor (AR10), and the bullet is stopped clean.