Calculating an impulse response in a stable environment
The easiest way to calculate the impulse response is to divide the spectrum of the measured signal by the spectrum of the excitation and do an inverse FFT of the quotient. That's mathematically equivalent to the cross-correlation method but typically easier. It requires that the excitation signal has a reasonable amount of energy at all frequencies so you don't get any "divide by 0" problems.

The length of the impulse response is mostly determined by the reverberation time of the room. For most of home environments that's well below 0.5 seconds so 1s excitation should be plenty. For convenience I would chose a power of two, which makes the FFTs easier. You can still truncate after the IR has been calculated.

Most acoustic systems are actually strict LTI (if no one moves about), however the signal to noise ratio (SNR) tends to be poor especially at low frequencies. A good way to deal with this is to do "coherent averaging". Create a periodic excitation (just keep repeating the sweep), throw away the first two frame acquisitions (long story) and then keep averaging the acquired frames. 10 averages will give you an SNR gain of 10 dB.

Getting the excitation level right is critical: if it's too low the SNR is bad and if it's too high, something will be overdriven and become non-linear. The speakers are probably the first candidate for distortion.