Freely suspended metallic single-wall carbon nanotubes (SWNTs) exhibit reduced current carrying ability compared to those lying on substrates, and striking negative differential conductance (NDC) at low electric fields. Theoretical analysis reveals significant self-heating effects including electron scattering by hot non-equilibrium optical phonons. Electron transport characteristics under strong self-heating are exploited for the first time to probe the thermal conductivity of individual SWNTs (~ 3600 Wm-1K-1 at T=300 K) up to ~700 K, and reveal a 1/T dependence expected for Umklapp phonon scattering at high temperatures.