Most swept-sine measurement systems (I have one that I compile in with any serious control code in my embedded stuff) will run a measurement unattended. Just leave a yellow card with "test in progress" on the thing and go have a nice lunch.
There's formal methods for extracting parameters from a step response, but if your accuracy need isn't super-stringent then eyeballing it with RC networks isn't a bad way to go.
Yes on the nonlinearity, although if it's not going to batter the power supply too much then a simple slow PWM to the heating element will take care of that right off, or use the pass transistor for the heating element (which is easy if you're talking about incubating eggs, but challenging if you're melting tungsten).