I have not seen much work on theoretical fundamental values of perpetual futures and how often the price deviates from them but there are some works on minimizing arbitrage gaps. While there might be slight differences amongst exchanges in terms of funding rate and funding interval, I guess they all rely on the same theoretical concept that is driven from traditional futures (the idea on how perpetuals should work etc.).
Funding Rate = Premium + Interest Rate
First the main purpose of Funding Rate is to try to converge the spot and perpetual prices (note that the price does not necessarily converge to the same value even at the end of the period and there are many reasons for that too but out of scope of this question). Exchanges have a clearly documented way on how they calculate the premium component. These calculation methods can slightly vary, and each method has some pros and cons (what is the fair time to calculate the rate, what should be the fair reference price etc. Even though those factors look trivial at first glance, when you have a millisecond perspective question will arise). Also, the interest rate component is different among exchanges (it is reflective of borrowing rate in that exchange). There are other things such as high trading costs that further create inefficiencies. This is just to highlight the small gaps that make things confusing.
To answer your question on why there’s a constant interest rate, its purpose is to reflect the borrowing rate of the underlaying (I guess your second point). Keeping other things constant you could go long on spot and borrow money and short on perpetuals. What’s happening is that you are pushing the price in both markets in the opposite direction. By assigning a constant rate this encourages someone else to take the same direction position in perps (hence try to converge the price).