Freely suspended single-walled 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 selfheating 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.