Uncertainties on the details of the Connor-Floyd isomorphism and Formal Group Laws

Question

Let $\Omega^{\bullet}(-)$ be the complex cobordism cohomology. $\Omega^n(X) = \{ (M, f) \mid f: M \to X \}$ where cobordant maps are identified, $M$ and $X$ are smooth manifolds, and $\dim(M ) = n$.

Should $\dim(M) = k-n$, where $\dim(X) = k$?

Let $x := c_1(L) \otimes 1$, $y := 1 \otimes c_1(L)$, where $L$ is the canonical line bundle

Each point in $\mathbb{CP}^n$ is a line through the origin in $\mathbb{C}^{n+1}$, which forms a line bundle which we call canonical? Why are we tensoring with 1 and what does this have to do with chern classes being stable?

It's my understanding that:

1. $\Omega^\bullet(-) \in \Omega^\bullet(pt)[[x,y]]$
2. $\Omega^\bullet(pt)[[x,y]] = \Omega^\bullet(\mathbb{CP}^\infty \times \mathbb{CP}^\infty)$
3. The homomorphism $\Omega^\bullet(-) \to K^\bullet(-)$ is of the form $\Omega^\bullet(-)\otimes_{\Omega^\bullet(pt)}K^\bullet(pt) \simeq K^\bullet(-)$
4. I've commented on the following diagram in the hopes that it will be corrected or clarified.