Clarification on how the emblem gained by Tibalt, Cosmic Impostor works
No and yes respectively to your questions.
For your first question, your analysis of rule 201.4a is correct, however you have an incorrect assumption in your question.  You assume that Tibalt's effect that creates the emblem can be countered separate of Tibalt, but it cannot.  Tibalt's effect is worded thus (emphasis mine):

As Tibalt enters the battlefield, you get an emblem with &quot;You may play
cards exiled with Tibalt, Cosmic Impostor, and you may spend mana as
though it were mana of any color to cast those spells.&quot;

Since it is worded &quot;As ...&quot; it is a replacement effect, not a triggered ability.

614.1c Effects that read “[This permanent] enters the battlefield with . . . ,” “As [this permanent] enters the battlefield . . . ,” or
“[This permanent] enters the battlefield as . . . “ are replacement
effects.

A triggered ability would be worded as &quot;When Tibalt enters the battlefield ...&quot;

603.6a Enters-the-battlefield abilities trigger when a permanent enters the battlefield. These are written, “When [this object] enters
the battlefield, . . . “ or “Whenever a [type] enters the battlefield,
. . .” Each time an event puts one or more permanents onto the
battlefield, all permanents on the battlefield (including the
newcomers) are checked for any enters-the-battlefield triggers that
match the event.

Since it is written as a replacement effect, there is no ability that ever goes on the stack that can be countered to stop creating Tibalt's emblem separate from countering Tibalt himself!  So you could never get into a situation where 201.4a could apply (under normal casting circumstances, copying Tibalt or some other jankiness can cause it to happen).  If Tibalt hits the battlefield, so does his emblem.
For the second situation, your analysis of rule 706.11 is correct.  The game will track Tibalt and any cards exiled by it, no matter if it's name changes or not.