For 1)
The functorial interpretation is developed by Strickland in
Formal schemes and formal groups. Homotopy invariant algebraic structures (Baltimore, MD, 1998), 263–352, Contemp. Math., 239, Amer. Math. Soc., Providence, RI, 1999.
an expanded version is on his webpage
https://neil-strickland.staff.shef.ac.uk/research/subgp.pdf
On quasi-coherent sheaves, the issue is certainly delicate, I would recommend you the initial sections in
"Duality and flat base change on formal schemes", which is the first paper in:
Alonso Tarrío, Leovigildo; Jeremías López, Ana; Lipman, Joseph:
Studies in duality on Noetherian formal schemes and non-Noetherian ordinary schemes. Contemporary Mathematics, 244. American Mathematical Society, Providence, RI, 1999.
For a readily available pdf (with some corrections incorporated)
http://www.math.purdue.edu/~lipman/papers/formal-duality.pdf
For 2)
The short answer is no, as far as I know.
For 3) there are famous counterexamples, see
Hironaka, Heisuke; Matsumura, Hideyuki:
Formal functions and formal embeddings. J. Math. Soc. Japan 20 1968, 52–82.
and
Hartshorne, Robin:
Ample subvarieties of algebraic varieties. Lecture Notes in Mathematics, Vol. 156 Springer-Verlag, Berlin-New York 1970
but also
Non-algebraizable Formal Scheme?
For 4)
As far as you stick to coherent sheaves, morphisms are automatically continuous, so invertible sheaves behave much as on ordinary schemes. Another completely different issue is ampleness, where counterexamples in hartshornes "algebraic Geometry" book show that the issue is subtle, and somehow connected to your second question.
Generalizing, you would like to seek for a nice system of generators of a substitute of the ill-behaved category of quasi-coherent sheaves. There are several possible candidates, but this line has not been pursued in these terms.
Best Answer
According to EGA I (Springer edition), Theoreme (10.10.2), there is an equivalence of categories between the category of finitely generated $A$-modules and the category of coherent $O_{{\rm Spf}(A)}$-modules. In particular, the answer to your question is Yes.