All characters and creatures are flat-footed until they've acted in combat. When the goblins act, bill still hasn't acted and so he is flat-footed.
A surprise round is a short round that's relevant if not all characters are aware of each other. If the players have listened at a door and know there are goblins inside, they are aware of their opponents. If they open the door, the goblins get a perception check each against an appropriate DC - those that succeed has noticed that the door is open. If the players act quickly, they can take advantage of the fact that not all goblins are aware, and so all characters, as well as the goblins that are aware, get to roll initiative and take a surprise round. They can only do a single move or standard action here though.
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Step 1) Determine who is aware that combat might be about to start.
Step 2) If everyone is aware that combat is about to start, skip step 3.
Step 3) Anyone who is aware gets a single standard action, in initiative order, before the first round starts.
Step 4) The first "real" round occurs, with everyone acting in initiative order as expected.
Note that those who have not yet acted are flat-footed. Acted first in the surprise round? Lucky you, no-one gets to count you as flat-footed. Acted last in the real round, and didn't participate in the surprise? Sucks to be you, especially if there're rogues.
To answer part of what the OP asked, "Flat-footed" doesn't mean you were surprised. It doesn't mean you failed to anticipate trouble. It doesn't mean you were cluelessly wandering in a goblin-infested dungeon and caught unaware when the goblins attack.
In fact, you can be unsruprised (you make your perception checks, or the attacking monster is noisy, or whatever) and yet if an enemy beats you on initiative, you are still flat-footed even though you know exactly what is going on.
Chalk it up to the time you spend figuring out what's going on and what you're going to do about it.
Holy crap! That's a lot of goblins! Where are they all coming from? Is it too many? Should I fight or run away? If I fight, do I have any chance of winning? If I run, do I have any chance of escaping? Ow! Crud! Those spears are SHARP!
OK, that's exaggerated, but that's really what's going on. The surprise round is quick (half a normal round, so you might call it just 3 seconds. If you're not surprised, you will get to act sometime in that 3 seconds. If you're not first, well, then you're still trying to assess the situation while some of the other combatants are getting busy with the wreckage.
And if you are surprised, and your enemies get a surprise round that means you spend 3 seconds basically in shock (surprised), and then during the normal round you are no longer surprised but might still need a little time to assess the situation (this would be the case if others beat your initiative) or not (which would be the case if you act first.
I understand the concept of warily exploring a cave and then being surprised by the inhabitants you know are there seems silly, but let me explain it at length...
Being flatfooted before you act in combat is like a reflex game. You know-- someone drops the yardstick past your hand unexpectedly, and you grab it as quick as possible. The marking where you grab it determines how quick your reflexes are.
Some people grab that yardstick quicker than others. Those people are able to react faster to someone "rolling initiative" on them. As a bonus, they're able to react so fast that someone who hasn't even reacted to their reaction can't react to their actions. Say that ten times fast.
Say I approach you in the park in the middle of the night. You can see me under the streetlight. I reach for my concealed pistol. You start running. I pull it free and open fire, missing you in the dark. You won initiative-- I was planning on attacking you, but you were able to both understand the situation, gauge it and act on your instincts before I could fire at you.
Now let's say I know you're going to attack me but even you don't know you're going to attack me. You might still get the drop on me. It might be one of those moments where I'm exhaling at the wrong second or for a split second I misjudge that I thought you were mind controlled. That kind of thing.
I'll just say that there's a reason why gunslingers who initiated duels were shot to death. They weren't quick enough.
Now, that's similar but different to the surprise round. Being flatfooted at the beginning of the round means you're still going to act that round, that you're completely aware combat has been initiated. So, what's a surprise round? A surprise round is when one or more parties aren't even aware that combat might happen. Let's go back to the park example:
Say I approach you in the park in the middle of the night. It's pitch black and you don't make your perception check to notice me. For all intents and purposes, I don't even exist to you. I reach for my pistol, pull it free and shoot you square in the back. Before I can get my second shot off, you're barreling through the woods in sheer terror. I fire, but miss. I got the surprise round-- you weren't aware combat was even going to happen, and so I got a whole standard action to myself-- but you still won initiative in the normal round.
On the flip side is the mind control example. I don't know you're mindcontrolled. You come up to me and your dominator is planning on attacking me when I turn my back. However, I notice something's wrong. I get a weird hunch when I talk to you, so I press it and realize you're under an enchantment spell. If you attack me now, I'm not surprised-- I have a reasonable suspicion that combat might be coming. If on the other hand I did not make my sense motive checks, I wouldn't know you were about to attack me, and you'd get a surprise round because I lacked that reasonable suspicion.
Dumbing it down a bit, an adventurer walks up to a door. The goblins are all asleep. The adventurer kicks down the door-- all of the goblins are surprised! Until they act, the goblins are flatfooted as well-- they didn't expect combat to happen!
Now the adventurer walks to the next door. The goblins inside heard the noise and are prepared for combat. The adventurer kicks open the door-- and loses initiative. All of the goblins act before he can, and the second the door opens, fill him with crossbow bolts.
But, before they can gloat too long, the ninja drops from the ceiling and forces another surprise round...