TL;DR: Although far from trivial, in a renormalizable theory, the diagrams with divergent subdiagrams and the diagrams with counterterms together yield finite correlator functions.   

Renormalization is a huge topic, but here are a few comments:

1. It should perhaps be stressed that OP's eqs. (1) and (2) both represent the same bare Lagrangian.

2. It is enough to consider connected propagators $G_c$ and amputated 1PI correlator functions $\Gamma_n$, because all correlator functions can be built as trees of vertices $\Gamma_n$ and lines $G_c$.

3. Renormalization of $\phi^4$ theory in 4D is discussed in many textbooks, see e.g. Refs. 1 & 2. In particular, the 6-point correlator function is mentioned in Fig. 9.1 of Ref. 2.


References: 

1. M.E. Peskin & D.V. Schroeder, _An Intro to QFT._ 1995; sections 10.2 + 10.5.

1. L.H. Ryder, _QFT,_ 2nd eds., 1996; sections 9.1 + 9.2 + 9.3.