Since you’re trying to flatten the phase from about 200 Hz upwards, I think it’s ok to leave the 100 Hz HPF engaged.
The bigger issue is the crossover and the relative physical placement of the drivers in the cabinet. Generally you can’t ignore these and expect to linearise the phase of the LF driver from 200 Hz up to crossover. The crossover filtering itself and the relative physical placement of the drivers both impart phase change through the crossover region. And so you need to measure the system as a whole. And like I said above, the manufacturer supplied crossover would have been designed with dispersion characteristics in mind.
The only time it may be possible to linearise the drivers separately is if you use a linear-phase brick-wall type crossover (like in the Lake processors). However you’ll likely still have a narrow frequency phase jump at crossover.
The 300 Hz versus 600 Hz issue could be that you’re designing the filter for 48 kHz sampling but the processor is running it at 96 kHz. Try changing the sample rate (in FIR Designer) to 96 kHz and see what happens.
Regarding question 3, FIR filters can be minimum phase (with no coefficients prior to the peak), linear phase (with coefficients symmetrically either side of the peak), maximum phase (with the peak at the end) or a mixture of all. Phase processing, as you are doing, generally results in a mixed phase filter. The “Filter delay” parameter enables the peak to be placed to minimise the error between the ideal FIR filter, and the filter windowed and truncated to the number of taps available (e.g. 1300). For more details, check out the links and tutorials referenced above.
When phase linearising using a FIR filter, the net effect is that the loudspeaker will have a bulk delay of approximately this “Filter delay” value. And if you are doing this on just one driver, the other driver will need to be delayed by the same amount. However see my previous comment about measuring the speaker as a whole.
If you measure the loudspeaker as a whole, you can still design the FIR for, say, 200 Hz to 1 kHz. The important part here is that you’ve measured WITH the crossover in the measurement/s.