Can Gate be cast directly under another creature to teleport them away?
Yes
The Gate spell is a Conjuration spell, but it's not a Teleportation effect. The subschool line of the spell tells us that it's a Creation or Calling effect. The Calling does not apply here, because the Gate wasn't used to call a creature, so this use of Gate was a Creation effect. As such this is not subject to the rules on the Teleportation subschool, but the Creation subschool.
The Gate spell itself does not specify the possible targets of the created gate, so we have to fall back on the general rules of the Creation subschool; and the first sentence there tells us all we need to know:
A creation spell manipulates matter to create an object or creature in the place the spellcaster designates. If the spell has a duration other than instantaneous […]
So yes, the Wizard can create the Gate underneath the Champion.
As for the created gate being a horizontal disk instead of the vertical one:
The gate itself is a circular hoop or disk from 5 to 20 feet in diameter (caster's choice) oriented in the direction you desire when it comes into existence (typically vertical and facing you). It is a two-dimensional window looking into the plane you specified when casting the spell, and anyone or anything that moves through is shunted instantly to the other side.
A gate has a front and a back. Creatures moving through the gate from the front are transported to the other plane; creatures moving through it from the back are not.
The spell makes it very clear that the caster decides about the gate's orientation. That it even says "typically vertical" proves that it doesn't have to be vertical.
As for being moved to the other plane:
Planar Travel: As a mode of planar travel, a gate spell functions much like a plane shift spell, except that the gate opens precisely at the point you desire (a creation effect). Deities and other beings who rule a planar realm can prevent a gate from opening in their presence or personal demesnes if they so desire. Travelers need not join hands with you—anyone who chooses to step through the portal is transported. A gate cannot be opened to another point on the same plane; the spell works only for interplanar travel.
The bolded part is not to suggest that you can only move through voluntarily, but is a modification to the mentioned spell Plane Shift, which has the following target line:
Target: creature touched, or up to eight willing creatures joining hands
The bolded part modifies this part, removing the necessity of the referenced spell Plane Shift. Gate itself uses this language to describe its travel:
[…] and anyone or anything that moves through is shunted instantly to the other side.
There is no reference to only those who willingly move through are shunted to the other side. This is why it also works with the Calling version of the Gate: The shunted creature need not willingly move through. However, since the travel works like Plane Shift, let's take a closer look at it again, especially this:
Saving Throw: Will negates
Spell Resistance: yes
Description: You move yourself or some other creature to another plane of existence or alternate dimension. If several willing persons link hands in a circle, as many as eight can be affected by the plane shift at the same time.
A creature can resist being moved either by succeeding on a Will save or the caster failing the spell resistance check on the creature. Spells that only affect willing creatures have the "(harmless)" addition in its Saving Throw and/or Spell Resistance lines. Plane Shift references "willing persons" only as another option of the spell: Either move 1 creature (willing or not) or move up to 8 willing creatures.
The Wizard was right that he could create a Gate in this way, but the Champion should have received a Will save (and spell resistance check) instead of a Reflex save.
