I can find no RAW, or any rulings at all, that can help answer this question. That being said, the following is speculation based on the entry on PHB page 205 that details ‘The Weave’ of magic.
In conjunction with how Wizards learn their magic, and the knowledge of the Arcane arts and the understanding they must have of the Weave in order to cast such magic, we can assume that yes: A wizard would know when the magic he cast a few turns ago that affected the Weave in such a way as to turn an ally invisible has lost its effect and thus become inert. I think this particularly holds true for concentration spells where the Wizard is actively maintaining the alteration to the Weave in order to sustain an effect.
When the spell ends, either through the Wizard deciding to end it, the spell running its full duration, or something ending it prematurely, I have no reason to believe this would be lost on the Wizard in question.
That depends on if it is casting a spell, or granting a spell-like-effect.
To cast an actual spell from an item, it requires your concentration. This is specified on page 141 of the DMG
[...] The spell uses its normal casting time, range, and duration, and the user of the item must concentrate if the spell requires concentration.
This rule would be used in the case of an item such as a Staff of Fire, which allows you to use it to cast the spell Wall of Fire (which requires concentration).
However, to cite the specific example you gave of a Ring of Invisibility. No.
Again, per the DMG, page 141...
Many items, such as potions, bypass the casting of a spell and confer the spell's effects.
The rule for concentration applies if the magic item allows you to cast a spell. It does not apply if the magic item grants you a spell-like effect. The stats on a Ring of Invisibility say
While wearing this ring, you can turn invisible as an action.
It does not say you can cast the spell Invisibility...just that you may turn Invisible as an action. This is an important distinction.
Thus, a Ring of Invisibility does NOT consume your concentration as it grants a spell-like effect, rather than allowing you to cast a spell. A Ring of Water Walking is another example of this, in that it allows you to walk on water at-will, rather than allowing you to cast the spell Water Walking.
Best Answer
A Potion of Flying specifically doesn't require concentration.
In page 141 of the DMG under "Spells", it specifically mentions potions as an exception to the rule of concentration:
So, if the potion does not say it requires concentration, it's safe to assume that it doesn't. The potions listed in the DMG all say in parenthesis that they do not require concentration but your DM may invent a house-ruled potion that does require concentration, so it's still safer to ask your DM.