Does 2NF require 1NF?
I will focus on questions 2 and 3, mainly by recalling a little bit of history of the Relational Data Model.

The first foundamental paper on the Relational Model was published in 1970 by the Turing Award Edgar F. Codd, Codd, E. F. “A Relational Model of Data for Large Shared Data Banks.” Communications of the ACM 13, no. 6 (June 1, 1970): 377–87. (bibliographic description, search pdf). 

In that paper it is expressed clearly that relations are sets:


  The term relation is used here in its accepted mathematical sense. Given sets S1, S2, ... , Sn, (not necessarily distinct), R is a relation on these n sets if it is a set of n-tuples each of which has its first element from S1, its second element from S2 , and so on ...


So, the requirement of the unicity of the tuples precedes each possible definition of normal form. 

Noting that the Codd’s definition leaves open the possibility that the elements of the domains (the Si) can be complex, for instance repeated, Codd introduced the first normal form (1NF) in a 1972 paper:  “E. F. Codd (Oct 1972), Further normalization of the database relational model, Courant Institute: Prentice-Hall, ISBN 013196741X,”. In such paper it is said that:


  A relation is in first normal form if it has the property that none of its domains has elements which are themselves sets.


As you have noted, this has nothing to do with functional dependencies, neither with other forms of dependencies (multivalued, join, etc.), and for this reason, it has been considered an essential requirement of the “normal” Relational Data Model. Other data models, the so called non-1NF model, abandon this requirement, and constitute a sort of bridge towards late object-oriented data model (but this is another subject...)

Only later, C.J. Date, a collaborator of Codd, wrote in the paper: “What First Normal Form Really Means,” in “Date on Database: Writings 2000-2006. Berkeley, Calif., Apress (2006)”:


  A table is in first normal form if and only if it is "isomorphic to some relation" ... There are no duplicate rows...."


The emphasis is mine, note that he’s talking about tables, not relations, specifying what are the characteristics that tables should have to be considered relations.

So, to summarize:


The relational model is based on the mathematical concept of relation, which is a set.
The 1NF has nothing to do with functional dependencies, and in the original definition has only to do with complex values (i.e. relations) inside tuples.
The other normal forms are defined ony in the context of the relational data model, in which relations are set.
Relational Database Management Systems (RDBMS) manage “tables”, which are loosely modeled on the mathematical concept of relation as defined in the Relational Data Model, but the normalization theory (functional dependencies, other kind of dependencies, normal forms, etc.) is only partially applicable to them, since it is defined only on the Relational Data Model.