Ad Q3: Chang's conjecture is preserved under c.c.c. forcing (your simplest case).

J. Barwise, P. Eklof, Infinitary properties of abelian torsion groups, Ann. Math. Logic, 1970, 25-68, is a classic reference (sciencedirect.com/science/article/pii/0003484370900069). L. Fuchs, Infinite Abelian Groups, Vol. I, 1970 (or Abelian Groups, 1960) is still a main reliable source.

@AsafKaragila But their long branches may be more visible on a sunny day.