In this work, we show that the nonlinear evolution of femtosecond seed pulses with different parameters (temporal and spectral shapes, repetition rate, pulse energy) in an Yb-fiber amplifier leads to gain-managed nonlinear amplification, enabling robust generation of high-peak-power and nearly transform-limited pulses after external compression. We demonstrate a compressed pulse duration of 33 fs with an energy of 80.5 nJ and a peak power of 2.29 MW for a source with a repetition rate of 30 MHz. For a second seed source with a repetition rate of 125 MHz, we obtained a pulse duration of 51 fs with an energy of 22.8 nJ and a peak power of 420 kW. Numerical simulations incorporating rate equations and nonlinear propagation in the amplifier provide evolutions that agree well with the experimental results. The discrepancies in the amplifier's absorption edge appearing at low repetition rates and higher pump powers are attributed to the temperature dependence of the amplifier's gain cross-sections. Here, we experimentally verify this attribution and thus underline the importance of accounting for the fiber core temperature for precise modelling of the short-wavelength spectral edge of the output pulses in nonlinear Yb-fiber amplifiers. We also measure, for the first time, the relative intensity noise of an amplifier operating in the gain-managed nonlinear regime. The measurements reveal a significant contribution of the amplification process to the overall output noise of the system.