Vibrational modes in suspended carbon nanotubes (CNTs) are incredibly soft, which makes them sensitive to small forces and prime candidates as force sensors. This same property, combined with the stiffness of the CNT to stretching, makes them an unusual mechanical system characterized both by large thermally-activated fluctuations and strong nonlinear interactions between the resonance modes. To understand how these thermal fluctuations manifest themselves in the resonance of CNTs, we developed an electrically-contacted micro-tweezer platform. The platform is capable of lifting a pristine CNT off of its growth substrate, directly applying strain to the free-standing doubly-clamped CNT, and controlling its proximity to electrical gates. Using the unprecedented level of measurement precision offered by our novel setup, we preformed the firstever single-shot ring-down measurements on CNTs, to map the resonance spectra as a function of strain and we directly measured the thermal motion of single-walled CNTs at room temperature. These measurements, in agreement with our original theoretical predictions, convincingly show that thermally-inducted fluctuations of CNT resonance modes are in fact the source of the remarkably high mechanical dissipation that has been ubiquitously observed in room-temperature CNT resonators. This result is not material dependent and the underlying physics should apply to all nanoscale 1D resonators. In addition to this key result, we use the microtweezer platform to couple CNT resonators to high-Q optical microdisk resonators. With this hybrid system, we demonstrate remarkably strong optomechanical coupling and make the first-ever observation of the optical spring-effect on CNT mechanical resonators.