In Lecture Notes in Algebraic Topology by Davis & Kirk, on page 240, there is written:
Q1: Is convergence of the filtration assumed in the first underline? Otherwise $\forall p: F_p=A$ is a filtration of $A$ with $Gr(A)=0$ despite $A\neq0$.
Q2: What is the precise statement of second underline? If $R$ is any commutative unital ring and $Gr(A)_p$ is free (of finite rank?) for every $p$ and the filtration is bounded above, then $Gr(A)\cong A$?
I think finite rank is necessary, but not sufficient: if $K$ is a field and $A\!=\!K^{(\mathbb{R})}$ and $F_n\!=\!K^{(\mathbb{R}\setminus\{1,\ldots,n\})}$, then $Gr(A)= \bigoplus_{n\in\mathbb{N}}\! \frac{K^{(\mathbb{R}\setminus\{1,\ldots,n\})}}{ K^{(\mathbb{R}\setminus\{1,\ldots,n+1\})}} \cong\bigoplus_{n\in\mathbb{N}}\!K =K^{(\mathbb{N})}\ncong A$.
What is the right formulation and how can it be proved?
Q3: In the third underline, what precisely is the statement? Ok, so there are weird filtrations, e.g. the $\mathbb{Z}$-modules $M\!=\!\mathbb{Z}$ and $M'\!=\!\mathbb{Z}\!\oplus\!\mathbb{Z}_2$ are not isomorphic and have different $2$-torsion, but there are filtrations $0\!\leq\!2\mathbb{Z}\!\leq\!M$ and $0\!\leq\!\mathbb{Z}\!\oplus\!0\!\leq\!M'$ with $Gr\,M\cong\mathbb{Z}\!\oplus\!\mathbb{Z}_2\cong Gr\,M'$.
But what does it mean that from $Gr(A)_p$ we can determine $A$ up to extension? That $Gr(A)\cong A\oplus B$ for some $B$ for which there is an exact sequence $0\rightarrow?\rightarrow B\rightarrow?\rightarrow0$?
For each graded piece you have a short exact sequence:
$0\rightarrow F_{p-1}\rightarrow F_p\rightarrow \operatorname{Gr}(A)_p\rightarrow 0.$
This gives an extension class in $\operatorname{Ext}^1(\operatorname{Gr}(A)_p,F_{p-1})$. Thus if $\operatorname{Gr}(A)_p$ is projective, then the whole Ext group is trivial, thus the short exact sequence splits and you get that $F_p=F_{p-1}\oplus \operatorname{Gr}(A)_p$.
If you glue these together (since $A=F_n$ for some $n$), you obtain:
$A=F_{n-1}\oplus \operatorname{Gr}(A)_n=F_{n-2}\oplus \operatorname{Gr}(A)_{n-1}\oplus \operatorname{Gr}(A)_n=\cdots$