Any carried object will be visible
Starting at 3rd level, when you cast Mage Hand, you can make the spectral hand invisible
"You can make the spectral hand invisible" does not mean "you can cast the Invisibility spell on it". It does what it says — you make the hand invisible, and only the hand, not the item it is carrying.
However, D&D 5th edition ...
A few, but they all explicitly require DM's permission
As it is a method of communication, can it be taken as a language?
Not as a part of standart character generation. You will require an explicit DM's permission:
With your DM's permission, you can instead choose a language from the Exotic Languages table or a secret language, such as thieves’ cant ...
It's a "secret language"
The rules divide languages into three groups: Standard, Exotic, and Secret. In the section on Languages about what you can pick as a language, the rules say:
Choose your languages from the Standard Languages table, or choose one that is common in your campaign. With your DM’s permission, you can instead choose a language from the ...
Yes, Fancy Footwork applies in these instances
Your reading is correct. To trigger Fancy Footwork all you need to do is:
make a melee attack against a creature
The cantrips Booming Blade and Green-Flame Blade use identical wording when they say:
you must make a melee attack with a weapon against one creature
So both Booming Blade and Green-Flame ...
Yes, rogues can evade AoE spells and other such effects
The term "attack" here isn't actually being used in any kind of special sense, and the ability should probably have been written to refer to "effects" instead of "attacks". Nonetheless, the intention is that if the rogue would be subject to any damaging effect which allows targets to make a reflex save ...
An incorporeal creature would be immune to the Sneak Attack dice.
The Incorporeal Subtype includes immunity to Precision damage except, specifically, when delivered with a Ghost Touch weapon; Sneak Attack is precision damage regardless of the type of damage it's amplifying.
You may be able to get around this with a ghost touch amulet of mighty fists; ask ...
Thieves' Cant is not mechanically equivalent to anything that can be swapped out, however...
True polymorph or Shapechange
The following monsters have Thieves' Cant listed for them:
Deep Scion (Volo's)
Master Thief (Volo's)
If you True polymorph or Shapechange into these creatures, you should be able to use Thieves' Cant.
Ring of Mind ...
The text in Cunning Action as written applies to both the turns a 17th level Rogue with Thief's Reflexes gets during the first round. If it said "once per round" rather than "on each of your turns", it would be unavailable on a Thief's second turn.