Not sure how you will go with Uber up to Mt Wellington but I believe they close the road to the top if there is snow or other dangerous conditions

There is a really good road going all the way to the summit with a huge carpark at the peak. Visibility on certain days will be your only issue. I can't see an issue.

No you do not.

You can drive, catch a bus or walk. The drive for the passenger's is stunning. The driver needs to concentrate as a couple of hairpin bends but from the bottom about a 20min drive. You can stop half way, park and walk to some amazing rock formations and views. Have been up in winter, very cold, need a down jacket, gloves, beanie etc. Even in summer usually cool and a breeze. Look from Hobart and if clear go for it. If cloudly wait until it clears.

The road is closed if the authorities think it is dangerous.

I would think not unless you could get a morning tour.

I doubt it, no.

I totally agree with you in that people shouldn't put a review out there when they haven't had a true experience of that place because of natural circumstances affecting their point of view. I was lucky enough to see the view and mountain on a clear day. I was told by my friends (who live there) that if it isn't a clear day you shouldn't attempt to go up there because the road will be blocked in certain places (depending on how good or bad the weather is). This is for everyone's safety. In general, I find some people will whinge about anything just because that is who they are. In my opinion, it is a beautiful part of the world regardless of whether you do or don't get to see it in it's full beauty.

No idea when it will snow, August is a cold month so hopefully but it’s all up to the weather

As per other answers if you drive carefully it should be safe to do so, weather permitting of course - Enjoy what a special view