Do you also have free weights? I believe if you have free weights (dumbbells and what not), then you can do the modifications to do the workouts.
For some things, you may be able to use your home gym but for other things you may need to use free weights.
Just thinking of the first phase, squats, lunges, deadlifts, stepups would require some free weights (or body weight). Lat pulldowns and rows could probably be done with your machine? Depending on your machine the shoulder presses could be done with your machine but since they are dumbbell shoulder presses, I'd say just use dumbbells. You'd need a ball for the 2 crunches.
As for how often, I'd say every other day although the book does give guidance on how often and I highly recommend reading the book through before starting the program.
You can't out-exercise poor eating habits.