References about Sierpinski's Theorem regarding Darboux functions
I followed advice given by N.S. and looked what A. M. Bruckner says about the origin of the first theorem from your question. (I still don't have anything to say about the second one.) However, I've noticed that you've found a different paper by Sierpiński which you mention in your paper.
Andrew M. Bruckner. Differentiation of Real Functions (LNM659, Springer, 1978). BTW there is also a newer edition of this book, but both editions say the same thing regarding the above theorem, see here.
Theorem 4.1. Let $f$ be an arbitrary function on $\mathbb R$. There exist two Darboux functions $g$ and $h$ such that $f=g+h$.
Theorem 4.2. Let $f$ be an arbitrary function on $\mathbb R$. There exists a sequence $\{f_n\}$ of Darboux functions converging pointwise to $f$.
Theorems 4.1 and 4.2 were first announced by Lindenbaum [121]. A clever proof can be found in Fast [59].
A. M. Bruckner and J. Ceder: On the Sum of Darboux Functions; Proceedings of the American Mathematical Society , Vol. 51, No. 1 (Aug., 1975), pp. 97-102 jstor.
In recent years a number of articles have dealt with questions concerning the possible outcomes of adding two real functions with the Darboux property (i.e. the intermediate value property). For example, Sierpiński [8] (see also Fast [4]) showed that in the absence of further conditions on the functions, every function is the sum of two such (Darboux) functions.
The articles referenced in the above excerpts are:
H. Fast, Une remarque sur la propriété de Weierstrasse, Colloq. Math. 7 (1959), 75-77. MR 22 #11087.
A. Lindenbaum, Annales de la Société Polonaise de Math. 6 (1927), 129.
W. Sierpiński, Sur une propriété de fonctions réelles quelconques, Matematiche (Catania) 8 (1953), no. 2, 43-48. MR 0064126.
Wikipedia provides a reference for the first theorem:
Bruckner, Andrew M: Differentiation of real functions, 2 ed, page 6, American Mathematical Society, 1994
Probably not the best approach, but you could check that book and try to backtrack the Theorem.