Let $X$ be a finite CW-complex such that its $K$-theory $K^\*(X)$ is, as a $\mathbb{Z}$-algebra, generated by $a\_1, \cdots, a\_n$ which are represented by reduced line bundles $L\_1-1, \cdots, L\_n-1$ satisfying $L\_i^{\otimes 2}\oplus 1\cong L\_i^{\oplus 2}$ (implying that $a\_i^2=0$) for $1\leq i\leq n$. Note that $a\_i$ may be a torsion element. We say $a\in K^{-1}(X)$ is represented by a reduced line bundle if, through the identification $K^{-1}(X)=\widetilde{K}^0(SX)$, $a$ is represented by a reduced line bundle on $SX$. Examples of such finite CW-complexes include $S^2$ and $\mathbb{RP}^2$, any product of them, and $S^2\times\_{\mathbb{Z}\_2}S^2$ where $\mathbb{Z}\_2$ acts on $S^2$ by antipodal map.


1. Is it true that the Atiyah-Hirzebruch spectral sequence for $K^\*(X)$ collapses on the $E\_2$-page, and there are no extension problems? It seems that $\text{Sq}^1$ and $\text{Sq}^2$ vanish on $X$, and $d\_3$, being the third integral Steenrod square, seems to vanish. I am not sure how to go about the higher differentials and the extension problems.
2. Is it true that the `Chern character map'
\begin{align\*}
\text{ch}: K^\*(X)&\to H^\*(X, \mathbb{Z})\\
a\_i&\mapsto \text{ch}(a\_i)=c\_1(L\_i)
\end{align\*}
is a ring isomorphism?


Note that the condition $L\_i^{\otimes 2}\oplus 1\cong L\_i^{\oplus 2}$ implies that $c\_1(L\_i)^2=0$, and it still makes sense to define the `Chern character' of any generator $\text{ch}(a\_i)=\text{ch}(L\_i)-1=e^{c\_1(L\_i)}-1=c\_1(L\_i)$ and product of generators
\begin{align\*}
\text{ch}(a\_ia\_j)&=\text{ch}(L\_i\otimes L\_j\oplus 1-(L\_i\oplus L\_j))\\
&=\left(1+c\_1(L\_i)+c\_1(L\_j)+\frac{(c\_1(L\_i)+c\_1(L\_j))^2}{2}+\cdots\right)+1-1-c\_1(L\_i)-1-c\_1(L\_j)\\
&=c\_1(L\_i)c\_1(L\_j)\\
&=\text{ch}(a\_i)\text{ch}(a\_j)
\end{align\*}
even for those $a\_i$ which are torsion $K$-theory classes. 



Atiyah-Hirzebruch spectral sequence for a special kind of CW-complexes
This is not true. One of the main reasons is that, because you've only specified properties about $K^\*(X)$, that leaves a lot of room for $H^\*(X;\Bbb Z)$ to have information which doesn't ultimately contribute to $K$-theory.


Finding an example is a little more work. The mod-2 Moore space $M(\Bbb Z/2, n)$ is the $(n-1)$-fold reduced suspension of $\Bbb RP^2$; it has the property that its reduced homology is $\Bbb Z/2$ in dimension $n$ and $0$ elsewhere. These have the property that for sufficiently large $n$, there is a map
$$
v: M(\Bbb Z/2, n+8) \to M(\Bbb Z/2, n)
$$
which induces an isomorphism
$$
v: K^\* M(\Bbb Z/2, n) \to K^\* M(\Bbb Z/2, n+8)
$$
on $K$-theory (real or complex $K$-theory, in fact). These maps were constructed by Adams in his study of the $J$-homomorphism ("On the groups J(X) IV").


As a result, we can take the mapping cone
$$
M(\Bbb Z/2, n+8) \to M(\Bbb Z/2, n) \to Cv
$$
and get a finite complex $Cv$ whose integral *cohomology* is $\Bbb Z/2$ in dimensions $(n+1)$ and $(n+10)$ and zero elsewhere; this means that the Atiyah-Hirzebruch spectral sequence starts nontrivially. However, the fact that $v$ was an isomorphism on $K$-theory means that $Cv$ has the same $K$-theory as a point. The space $Cv$ then satisfies your criteria for not very interesting reasons. 


The fact that $Cv$ has no $K$-theory means that the Atiyah-Hirzebruch spectral sequence has to erase everything except the $0$-line (in fact, there is a $d\_9$ differential). This also means that the Chern character doesn't have a hope of being an isomorphism because the source ring has no torsion and the target does.


