1) I'm not so certain it's realistic to operate that fund and see a cash flow that you can take out of the business as a profit on the part-time. Being a landlord is so frequently a full-time job, especially if you have 4+ units. If you've only got the three days a week to participate, you're going to need to have an awesome team.
2) I actually love the idea to come into someone else's deal as an LP. Getting to know the business model, seeing what people do with their money, and how they make money is a lot easier than trying to make your own deal work for the first deal. I think learning the business model first is great, especially if you haven't already targeted property for acquisition.
3) I'd start with the CPA because the money is always going to be the first thing on your mind as an investor. I would then hire a property manager and ask both the CPA and the property manager for a recommendation on a landlord-tenant attorney. You probably won't need a full-fledged real estate attorney unless you're developing the project or going for rezoning. Just a local firm that knows the ropes and the game is definitely good enough for your model.
4) Who are you looking to communicate the deal to, your investors or the sellers? If you're having difficulty getting the deal correctly worded for your investors, maybe hire the law firm first so that everyone feels like they're properly protected. It's so important to get these things started with harmony in management.
5) Think about your exit strategy. I tell my investors this all the time, I want them to think under what circumstances they'll sell and look to move on to a new project. If you know you're take-and-hold as a strategy, then you know that you're gonna have to budget reserves for long term capital improvements. If you know that you're 3 years from flipping, then you hold enough for 3 years expenses. You need to know how long to visualize trends in the area and what kind of variance to predict. I would also tell you to really look into how you're going to keep these units occupied as apartments in the rural south. Rural tenancies can be difficult because there aren't often many young people looking to get started on moving into their first place. They can also have a strange habit of staying vacant during the low-times as they're not so near development and jobs.
I also think your purchase price numbers are a bit high variance. You want $1mm-$4mm, which is a huge spread. A $1mm building (in my area) is a 6 unit in good condition that is ready to rent. A $4mm building in my area is a 20 unit or a 24 unit. A $1mm building is something you can do yourself if you're handy and self-motivated, a $4mm building is something you definitely need staff for. I believe it might be best to narrow your search criteria a little because you're really looking at multiple different REI opportunities of immensely varying complexity given the price points you've named.
I do think you've got a solid starter plan, I hope you can realize it and succeed in 2020 and beyond!