Regarding first question:
To be sure, I checked Modern Physical Organic Chemistry, which says "Atoms with higher polarizability form the incipient covalent bond to the electrophile more easily because its electron cloud more readily accommodates changes in shape."
In the larger/more polarizable atom, the valence electrons, which are used to form bonds, are further from the nucleus and there are more core electrons shielding the valence electrons from the positive charge of the nucleus. So they are more easily pulled toward another nucleus to form a bond.
Polarizability is something that may correlate with higher Nucleophilicity. Relative nucleophilicity, especially for anions, often depends on the solvent. There are many articles online about solvent effects on nucleophilicity. Polar aprotic vs. polar protic is the main thing.
Regarding second question:
Basicity is a thermodynamic concept, so it refers to the relative stability of the acid and base form. A base is strong because it's an unstable anion and/or because it forms a strong covalent bond to hydrogen. Generally it's some combo of those two things, although some neutral bases containing nitrogen can be rather strong also.
Polarizable atoms tend to be large. Large atom monatomic anions are supposedly stable, I guess because the negative charge is spread out. Just as importantly, large atoms also make weak bonds to hydrogen, because the orbital overlap is bad between the 1S hydrogen orbital and the higher energy orbitals of the large atom.