I just wanted to mention another class of residually finite groups that have lots of strange properties, even though probably Yiftach knows about them.
This is the class of Generalised Golod-Shafarevich (GGS) groups. These are groups with "sparse enough relations", so that the group can be at the same time be small and have lots of quotients. They were used by Misha Ershov and Andrei Jaikin to solve many open questions on abstract and pro-p groups.
A few properties, a f.g. GGS group: has exponential word growth, has fast subgroup growth, has a GGS quotient with property (T), has a GGS torsion quotient, has a GGS hereditarily just infinite torsion quotient.
About finite presentation: these groups can be finitely presented, but it is conjectured that a GGS f.p. abstract group contains a free subgroup.