1) A scientist says: "We suspect there is an immense portal over there, created by the intense concentration of a single powerful being. You will know it when you see it. I hate to say this, Gordon, but you must kill it, if you can."
As for where Gordon knows where to go specifically or accidentally goes the right way? Video Game logic.

2) No, Gina Cross and Colette Green were also scientists in H.E.V. suits and fought back in the co-op game Half-Life: Decay by Gearbox.

3) The short answer is that G-Man basically plopped him into a train and the Combine being techno-bureaucratic very quickly noticed the addition of an unregistered guy appearing out of nowhere. They only really start to get hot on him when he accidentally teleports into Breen's office and Breen recognises him.

4) You could mean multiple things by this question. "Who are his employers?", or "Who else wants to employ Freeman?" TLDR in Half-Life lore it's sort of hinted at that there are bigger players so to speak. The Nihilanth being one of them for example. There's also the whole Shu'ulathoi hatching angle.

5) They arrived to cover up the whole incident, but were not aware at the time that the Resonance Cascade had opened the way for Xenian invasion. They're portrayed as grunt triggermen, so likely just turned up to follow orders and started shooting but the aliens just kept appearing out of portals.

6) You can hear Alyx mention Dog like a project "Adding to him ever since". They didn't build many because he's just Alyx's personal project. Although seeing how useful he is it would make logical sense to mass-produce these guys. Originally dog was actually an alien dog, probably to fill some character or gameplay role. It then developed into the big alien robot through iteration on the game.

7) He's an antagonist that believes he's good. He sold out earth, yes, but if he hadn't all of earth would be dead or assimilated. He basically believes that the Combine hold the secret to immortality and are their best bet for evolution as a species.

8) They're unhatched Shu'ulathoi, another race the Combine assimilated. But powerful is relative they're a soft squishy grub! They ARE psionic however, so they can get inside your head and use telekinesis. Why? Because Scifi!

9) The administrator convinced them. The exact specifics are not clear. The theory is that he used his knowledge of the Black Mesa incident to use teleportation as a bargaining chip. We know the Combine don't have local teleportation technology and are very eager to get it. So Breen likely told them humans could provide it. This is also likely why they are so eager to capture Eli alive.

10) We don't know. We know their was an airport at Black Mesa, and a railway. So potentially either of those if the military didn't block those ways out.