Let's put aside any blind and uninformed hate of insurance that's seen in threads like this. Get to the facts.
First, do you want to file with the other carrier or your own? That's a decision you have to make. You really don't want to do both simultaneously, it'll just lead to confusion and probably slow the process down. I realize that ultimately you want the other drivers' insurance to pay for it, and at the end of the day, as long as coverage is applicable, they will. However, do you want to deal with them directly now or go to your own carrier and let them deal with it? Make that decision now and stick to it. Given that there are two totaled vehicles (and yes, I think your Yukon is totaled), I'd suggest going to your own carrier. I'd be concerned that there isn't enough insurance coverage through the other driver to cover both cars. Maybe there is, maybe there isn't. If you go to your own carrier, than it's no longer your problem, it's theirs. The role your company plays in it is if you file with your own company, they will pay your damages (less deductible) and then pursue the other company to get all the money back, including your deductible. If you file with the other carrier, your insurance doesn't really play any kind of role. As for your question about the rental, the other carrier likely has either a coverage issue or simply needs to talk to their own driver. Until either or both of those are cleared up, they're not going to officially accept liability and pay for a rental. Liability seems pretty cut and dry, so my guess is either it's a coverage issue, a policy limits issue, or they simply haven't spoken to their driver to confirm facts yet.
Either way, honestly, don't rely on your agent. That's not what they do. They are primarily salesman/women. Their job is to determine the type of coverage you need and find the best way to get you that coverage. Claims is typically not their expertise. Some are better than others, but I can't tell you how many times I've seen it go sideways. They mean well, but often don't really know what they're doing. Deal with the claim departments directly.
No offense whatsoever, as I know it's very personal for you (as it rightly should be), but there really isn't anything "special" about this case to the insurance company. Your car was parked and unoccupied when it was hit. Coverage or limits issue aside, it's really a "Claims 101" type loss for the carrier...about as basic as it gets. Don't expect anything other than the value of the Yukon. On that note, yes, they only owe you cash value of it. If there were anything "special" about your loss, that would be it...you had owned the vehicle for a matter of hours. So if I were the other carrier, or even your own, unless you drastically overpaid for it (or underpaid), I would be inclined to give you exactly what you paid...as opposed to the normal handling, where you are given actual cash value.
Hope this helps. Let me know if you have any questions.