Is there a difference between "cell line" and "cell strain"?
"Strain", in general, is usually used for whole organisms (whether they are unicellular or multicellular). For example, there are mouse strains.

"Cell line", on the other hand, is a very specific term that is used to refer cultured cells obtained from a multicellular organism. They need not be necessarily aneuploid.


  Is a cell line always aneuploid and capable of indefinite growth, as
  opposed to being euploid and incapable of indefinite growth?


Aneuploidy does not have any causative relationship with indefinite growth. Most cancerous cells evade the cell cycle checkpoints, which allows them to grow out of control. Some of these checkpoints also ensure proper segregation of chromosomes. Bypassing them leads to aneuploidy.


  Are cell lines essentially immortal?


Usually primary cell cultures are not referred to as "cell lines" these days. However, originally, any cultured cell that grows in vitro is called cell line. From Freshney (Basic Principles of Cell Culture):


  When cells are isolated from donor tissue, they may be maintained in
  a number of different ways. A simple small fragment of tissue that
  adheres to the growth surface, either spontaneously or aided by
  mechanical means, a plasma clot, or an extracellular matrix
  constituent, such as collagen, will usually give rise to an outgrowth
  of cells. This type of culture is known as a primary explant, and the
  cells migrating out are known as the outgrowth (Figs. 1.1, 1.2, See
  Color Plate 1). Cells in the outgrowth are selected, in the first
  instance, by their ability to migrate from the explant and
  subsequently, if subcultured, by their ability to proliferate. When a tissue sample is disaggregated, either mechanically or enzymatically
  (See Fig. 1.1), the suspension of cells and small aggregates that is
  generated will contain a proportion of cells capable of attachment
  to a solid substrate, forming a monolayer. Those cells within the monolayer that are capable of proliferation will then be selected at
  the first subculture and, as with the outgrowth from a primary
  explant, may give rise to a cell line.


Elsewhere, a cell strain is defined as positively selected subpopulation of cells:


  A cell strain is a subpopulation of a cell line that has been positively selected from the culture, by cloning or some other method.  A cell strain will often have undergone additional genetic changes since the initiation of the parent line. Individual cell strains may, for example, have become more or less tumorigenic than the established line, or they may be designated as a separate strain following transfection procedures.


This definition does not say anything about what kind of selection was used. You can select for cell that express GFP, for instance. So that strain would be simply designated as GFP+.