Regarding your last paragraph: The proof of the Ternary Goldbach Conjecture proceeded in two steps: 1) prove the result for all numbers bigger than a certain, known number $n$, and 2) use a computer to verify the result for all numbers less than $n$. It is conceivable that a similar method of proof would work for the Riemann hypothesis.

Who is the "R" in BNSR? (I know BNS=Bieri, Neumann and Strebel.)

(...and contains many, many errors...)

@NoamKolodner Lyndon and Schupp has not been updated since its original publication.

@HJRW The arxiv version of the second paper has the sentence "The counter part of the Seifert-van-Kampen Theorem in geometric group theory is Stallings’ structure theorem". (The arXiv version is 3 pages longer than the journal version and seems to contain more background.) [For completeness: The first paper has the sentence "Stalling’s [sic] Theorem...is the analogue of Seifert-van-Kampen’s Theorem in geometric group theory." ]

A result of Ashot Minasyan goes some way to giving a negative answer to your question, at least for countable groups: Theorem. Let $C$ be an arbitrary countable group. Then for every non-elementary torsion-free word hyperbolic group $H$ there exists a torsion-free group $N$ satisfying the following properties: 1. $N$ is a $2$-generated quotient of $H$; 2. $N$ has two conjugacy classes; 3. $\operatorname{Out}(N)\cong C$. See Theorem 1.5 of Minasyan, A. "Groups with finitely many conjugacy classes and their automorphisms." Comm. Math. Hel. (2009) MR2495795 arxiv.org/abs/0704.0091

Okay, thanks. I'll do the exercise :-).

For (5), $\mathbb{Z}$ works. I think also any cyclic group of odd (prime?) order. The idea is to note that $G/Z(G)$ embeds into $\operatorname{Aut}(G)$, and hence $G$ is abelian (standard undergrad exercise). Such a group always has an automorphism of even order, assuming AOC (less standard exercise, not sure of level), a contradiction.

Somewhat related to Adam P. Goucher's answer: your group is a 2-generator, 1-relator group with torsion, and such a group $G=\langle a, b; R^n\rangle$, $n>1$ is Fuchsian if and only if $G\cong \langle a, b; [a, b]^n\rangle$ (see Fine and Rosenberger, Classification of all generating pairs of two generator Fuchsian groups (Groups St Andrews, 1993), if and only if $R$ is a cyclic shift of $[a, b]^{\pm1}$ (from the isomorphism problem for 2-generator, 1-relator groups with torsion). So your group is Fuchsian, which is essentially what the answer notes (my point is that it generalises to $n>2$).