I have an X-300 with over 1,000 hours. It was not properly maintained by the PO. One of the first things I did was change hydro oil. That means dropping it, inverting it, draining, the turning back upright, filling, and reinstalling. Alternately, you can remove, take the bottom plate off and drain that way. Not something most people want to do yearly. There are instructions out there on how/where to drill holes and install drain plugs, but I didn't do that. It helped, but it's still weak, moreso once it gets hot.
As it was explained to me elsewhere, that level of machine is a lawn cutter, intended for urban or suburban lawns that are reasonably level. They are not intended for work. They are not intended for steeper slopes. They are not intended to pull things other than wheeled carts.
OP needs to be looking for a GT, Garden Tractor. They have more substantial transmissions and are designed to do more of the tasks outlined.
Yes, you want a Hydro, unless you want to be shifting gears every few minutes doing routine tasks.
All hydros 'wheeze'. That's the sound of fluid being moved under high pressure. Some do it louder than others. Some move uphill slower than others, but are capable of pulling more weight.
Locking differentials will help in situations where one wheel spins. I'm not sure it's a sign of a higher end transmission.
I've seen cases where the wheel bolts on to a hub, but the hub slips onto a shaft with a key, so no, not necessarily.
Do research, check specs, avoid lower ends transmissions for what you want to do. Look for a GT, Garden tractor, not an L (lawn) or similar. The lower end Big Box store riders may have an E or D instead of an L. A Garden Tractor will likely be $2500 minimum and can go up to $7500 or more. A $1500-$2000 rider will have a lower end transmission.
As far as any kind of bucket on the front, you need to be concerned about front suspension and how the wheels are mounted. Lower end grass cutters have a simple shaft, bushing and pin type mount with a tie rod for steering control. Also, the front portions of the frames are usually stamped and folded sheet metal. They will not handle a loaded bucket well. .
There has been some discussion here about a Husqvarna model but I can't remember the number, something 354 I think.. Also the Cub XT3 has been mentioned.