Give the dealer another shot to get it right; if you don't have confidence in their work, there are other dealers in your area. Here's a link to search by ZIP code:
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Should not squeak; the belt engage process varies depending on the version of the VKA you have, some have a stiff spring that holds tension, while others tighten/loosen when you engage the control.
Unless there's evidence of leaking fluid, no reason to check the fluid level. The transmission uses the same oil as the engine, SAE 10W-30, and the transmission must be removed from the mower to open it and check the level. Again, no need to do this unless there's evidence to justify the work. The oil in the transmission will last for the life of the mower.
Here's how the front wheels come apart; since you're still under warranty, discuss possible repairs with the dealer, but BE SURE to confirm if any needed repairs will be covered or NOT covered under the factory warranty before authorizing the work to be done:
Use general-purpose NGLI #2 grease where shown. Confirm the curved wheel washer is installed properly. The wheel bolt needs a bit of grease, but don't overdo it or overtighten.