Let's look at each item, and how it affects Gangplank:
Philosopher's Stone - this is a strong item on many champions. It gives him the mana regen he needs to use Parrrley, HP regen, and GP/5.
Wriggle's Lantern - provides armor, AD, life steal, a unique passive and a free ward. This is another strong pick, especially if jungling Gangplank.
Mercury Treads - these boots are considered the default tank boots, due to MR and tenacity.
Spirit Visage - provides health, magic resist, CDR, and a boost to healing. I don't recommend this item because it doesn't synergize well with any of his abilities.
Frozen Heart - provides armor, mana, CDR, and an aura debuff. This item is best vs an auto-attack heavy team, otherwise I would skip it.
Thornmail - provides armor, and deals magic damage when auto-attacked. I don't recommend this item on Gangplank, because a clever opponent will stop focusing Gangplank when they see you've purchased it. Since your goal is to be a tank, this is counter-productive.
Warmogs Armor - provides the most health in any item, and good health regen. Helpful for staying alive.
Atma's Impaler - provides 15 AD for every 1k HP, armor, and crit chance. A solid pick if you have extra health from runes/items.
The first build you've listed is very defensive. You end with 229 armor and 55 MR. This build would be most effective against a team that relies on auto-attacks (Master Yi, Ashe, etc). It's going to be weaker vs a team doing primarily magic damage. Warmog's and Atma's provides more health (good if you have armor/MR), and some damage, which helps
If you're jungling Gangplank, it's common to rush a Wriggle's first. If you're top lane,
you're going to focus on spamming Parrrley instead, so I'd recommend rushing a Philosopher's stone. Sheen is a common pick, because Parrrley trigger's Sheen's bonus damage. It also builds into a Triforce, an item that boosts all of Gangplank's abilities.
Every game is different, and you have to build vs your enemy team. Of all the items listed, Mercury Treads and a Philosopher's stone are always viable. Wriggle's is very common to see on him, and most of the time will be a good pick. From there, build a sheen if you're using Parrrley a lot, or health, armor, and/or magic resist depending on what's doing the most damage to you.
tl;dr: Mercury Treads, Philosopher's Stone, and Wriggle's Lantern are almost always good picks. Everything else is situational.
The primary reason is for Wave Clear.
Waveclear is important for controlling your lane, and for pushing it to roam. Fizz is also quite a poor last hitter under turret, so any mid laner that can easily shove him in will cause problems.
With that said, the Cooldown is also significantly decreased with level, and as it is Fizz's primary source of safety, having that as often as possible is nice.
It changes the way you attack targets though, you can't all-in in quite the same way as if you max W or Q. Basically you want to land the E first, then Q back through towards the safe side with some W procs along the way. Then wait till your cooldowns are back and do it again.
As for the W last. It does increase it's damage a decent amount with level, but again this is a matter of reducing the cooldowns. Increasing your passive damage by 1% per level really isnt that much (with W), whereas decreasing the cooldown of your Q by a few seconds is much more significant for rotating through your spells again, and for chasing in general.
Best Answer
Yes, all those items are able to proc the effect of windspeakers blessing.
Edit 14.06.16: While i previously stated that Galios Bulwark does proc windspeakers blessing, this is only true for the heal. It does not grant a shield, but only armor/mr, which is not affected by windspeakers blessing as far as i know - I'll test it today when i come home and edit the result in here.