The change to the Rivian website in which "you pays your money and ya takes your chances" in terms of actually choosing a configuration is very new, and I don't think anyone at this point knows how it works, let alone delivery timing. The ordering and delivery process at Rivian has always been inscrutable (something Rivian could easily fix but appears to have chosen not to) but is now, as outlined below, nearly completely dysfunctional.
Most (maybe all) folks who put pre-orders in place before 2022 got emails last December with their "delivery windows". Unfortunately, Rivian has had to revise those projected delivery dates for all but a relative handful of lucky people. To make matters worse, Rivian no longer appears to be fulfilling orders with much concern as to when they were made. As you may be aware, there are increasing, credible reports of folks who ordered only a few months (in one recent case, only a few weeks) ago getting their vehicles. The reasons for this are not apparent, and Rivian has not seen fit to provide any clarifying information, but it has, unsurprisingly, had the effect of really pissing off folks who've been waiting 2 and even 3 years for their vehicles. The process has come to be called - appropriately in my view - the Rivian Lottery.
I don't know what your odds of winning the Rivian Lottery are, but they're apparently not zero, so you could see an R1T in driveway in a few months. More likely however, you will be nearer the back of a 90,000-plus queue, which will in turn mean a substantial wait. It's really tough to say how long of a wait, but I'm reasonably certain it will not be as long as the 3 years mine will have been if the 1H2023 delivery window that Rivian provided is accurate (unlikely, my order is for a Max, so 2024 is a much better bet, putting the wait at 4 years.)
As to the dog issue, we downsized when our longtime pair (Dane and Setter), passed on after 12-1/2 and 15 years respectively, and now cart around two incredibly animated Welsh Springers. We have an SUV and use seat and cargo covers, along with moving blankets, to help protect the interior; the capacity for Welshies to find and retain copious amounts of grit, mud and flora is unmatched in our canine experience.