I currently have my Tesla Powerwalls set up to provide full house backup except for HVAC. I'm in Los Angeles, so I don't have truly cold weather to deal with in the winter, and AC in the summer would deplete the backup power too quickly. If I add two more Powerwalls I'd probably reconfigure to house backup to include HVAC. My two 240v Level 2 car chargers are also not wired to the Tesla backup system. That's an LADWP thing -- I must get at minimum a 3rd Powerwall to include that. However, whenever I top-up charge from 120v Level1, I am charging from solar or battery-stored solar now. In addition to backup, the Tesla Powerwalls power the house after sundown or sometimes during the day if it's raining or unusually dark cloud conditions. The software dynamically manages power sourcing and distribution, splitting solar power between home and Powerwalls, switching the grid incoming power off and on as needed, etc. You can monitor powerflow via an app on your phone or tablet. If the grid fails, switchover to backup power is usually undetectable. Sometimes there's a flicker but usually not.
Whether a Tesla installation saves you money over PV alone is a matter of studying several variables that are regionally or municipally influenced, On balance, selling daytime excess PV power back to the power company real-time and consuming from the grid in the evenings seems a little more expensive than capturing the excess PV to consume myself in the evening. But that may not be true for you. I would have more panels installed but local power company regulations proscribe how much PV capacity can be installed, based on prior year's power consumption. Some people plan ahead and intentionally jack up their electricity consumption for a year prior to getting a PV installation, to get cleared for more generating capacity. If you are truly going off the grid, you can sidestep this, but if you are remaining connected to the grid, then the power company has approval rights for how much capacity gets connected to their grid. Point is, if I had enough panels to be sure of generating all the power I would need to in the winter, my summer surplus would be more than I can consume (if stored) or offset through sell-back. I'd be a profit-making enterprise probably 8 months of the year, and the power company won't let me do that.
During the day, unless it's a winter's day of rain, I am running on PV. That doesn't make my hifi sound better nor worse than grid. In the evening when I am powered from the Tesla batteries, sonic effects are not compelling and are often nil. But I sometimes can convince myself the batteries can pull that midnight "quiet grid" sound earlier into the evening. I think the reason there aren't more obvious sonic benefits is that in both cases, you have DC systems that have to power your AC house via inverters. There are probably some choices one can make among inverters, but the electricians and vendors involved are, in my experience, completely inexperienced (and uninterested) about the hifi implications of their gear and installations. So I'd have low expectations in that respect, and be grateful for any surprise benefits.
Powerwall installations allow you to configure through the Tesla app, how much capacity to reserve for backup when in self-power mode. If you set for 100% backup or backup-only mode, you will only draw from the batteries when the grid fails. In my area, if you have taken rebates, you are limited in how much capacity you can reserve for backup power for the first two years, and cannot run Powerwalls in backup-only mode. If you have solar installed, then you will draw directly from your PV panels and some of that power will also be diverted to recharging the Powerwalls. You will draw from the batteries only when you don't generate enough PV power for your current needs. The system will blend in battery power as PV output fades and falls below your consumption. And when the batteries are depleted, it will revert to grid power. If you do not have solar and have installed batteries as backup against grid failure, you will pull from batteries only when it makes economic sense, if you allocate less than 100% capacity to backup.
Point is, it's a smart system, and you don't get an override to force consumption from batteries based on your hifi habits, if they don't coincide with the system's decisions about when to charge, when to wait, and when to release battery-stored power.
Phil