Anyone has advantage when doing this?
Anyone can make a Str(Athletics) or Dex(Acrobatics) "attempt". If an anchor point is available and the DM judges that the action being attempted would benefit from the presence of a rope or similar aid to climbing/swinging etc, granting advantage on the check would be reasonable.
The presence of a feat that grants this specific ability however, would imply that you can't do it without that feat.
A grapple says "Using a free hand" and don't see any exception from whips.
A shove is more controversial since it uses the phrase the creature "must be within your reach."
But a reach weapon "adds 5 feet to your reach when you attack with it," and shoving is considered a "special melee attack."
So you're using a melee weapon that grants reach when you attack with it and you're making a special melee attack. Not really seeing a problem.
Rewarding the free hand for grappling, if we have to get that picky, does it specify that you have to touch the target, or could you use the free hand to help control the whip for example?
Oh, sure, a generous DM might allow you to do this even without the feat. But the DM might just as likely never allow you to. The Feat would be a nudge to the DM to be generous.
The feat would be a nudge to the DM to
not be generous: to
not allow advantage for using a whip or rope or such on Athletics or similar checks
unless you had the feat.
On a more general note, your logic would suggest that no mechanic should ever contain the phrase "at the DM's discretion" because everything is already at the DM's discretion. Doesn't the wording for the Hide action say something about this (or is that only the Sage Advice clarification?). So...to be a bit facetious for a moment...maybe we should just take Hide out of the rules since clearly DMs already have the discretion to allow characters to hide.
How is that
my logic? You're the one who wrote that phrase, not me.
Now if you want to get more specific than "Strength (Athletics) or Dexterity (Acrobatics) attempts" we can start specifying examples where the whip would only be useful if you had unusual training in it perhaps.
Link to the YouTube video of you doing this?
I would most certainly not be considered to have proficiency in Whip, but if you can point out somewhere suitable near me and lend me a whip, I'm pretty sure that I could demonstrate how using a long, flexible tool makes some athletic tasks easier.
As Tusk points out, using a reach weapon doesn't extend YOUR reach, just the weapon's.
You're using the weapon to make the attack.