Not a PT, but I've been training in the gym for about a decade, so take this anecdotal evidence with a grain of salt.
Some lower back soreness is acceptable, but still not a great thing. Your core should be strong enough to keep your torso rigid without having to load weight on your lumbar. Try using a lifting belt to help with your bracing. Are you low-bar or high-bar squatting? That will greatly change the position of your torso when you squat down. High bar is more upright and shouldn't involve your back as much.
If your gym has one, try a safety bar squat. I use that more than a regular barbell now because I have had back issues in the past and I still have shoulder issues that make squatting heavy weight with a barbell hurt my shoulders. The safety bar moves the weight for ward and makes the movement almost like a front squat, really targeting the quads but it keeps your torso almost totally upright and should take your lower back out of the equation.
Not sure why slant boards are causing knee pain, possibly they are causing you to shift your weight and balance too far forward in the squat. I know that when I used squat shoes, I found myself getting pressure and pain in my toes from the weight squishing them into the front of the shoes but no knee pain. I stopped using squat shoes and just focused on form over weight using flat shoes (converse, wrestling shoes, barefoot shoes). Knees-over-toes is acceptable and even good form when you're squatting, but you should still try to have your balance directly over the middle of your foot and drive through your heels using your hips.
I also personally think that hips parallel with knees is deep enough, and ass-to-grass is only good when you're flexible enough to get that low without lower back extension. For a lot of newbies I tell them to only squat as low as they can get before the "butt wink." I also still don't have great ankle flexibility, it's a consequence of our sedentary lifestyle and using chairs vs squatting to rest. The flexibility will come with time, and stretching your calves and Achilles tendon is what we need to work on to improve ankle mobility. I know for me, my calves are tight as hell and it's something I need to be working on more too.
I'm sure you know this already but youtube is a huge resource for lifting knowledge. Squat University is a great channel run by a physical therapist and he has a video on almost every topic.
(Sorry for the essay but I love talking about weightlifting haha)