You have the formula for dividend yield of a mutual fund correct. It is the annual dividend payment divided by the value of the mutual fund.
This metric is just a snapshot of what the mutual fund is currently paying out at the time you calculate it. The net asset value (NAV) of a mutual fund changes daily as the price of the stocks contained inside it change, and the dividend payment is different everytime it is paid as well, since there are many stocks in the mutual fund that pay dividends.
Discovery 1 - The fund has a fixed-dollar dividend $0.3 per share, not a fixed-rate (3%) dividend. (Is fixed-dollar dividend common, or is fixed-rate dividend common?)
Yes, dividends for mutual funds are typically declared per-share, just as the dividends for individual stocks are declared. The price of a stock goes up and down continuously while the market is open, but when the dividend is announced, the payment is fixed no matter what the stock price happens to be doing on the day it is paid.
Discovery 2 - If the share price drops to $6, the dividend stays the same at $0.3 (unless a change is declared), but the yield is now 0.3/6=5%.
Sure, a change in value affects the yield calculation, just as you would expect it to. As I said, the yield calculation is simply a snapshot of the current situation, and does not take into account what you originally paid for the fund or any growth or loss in the value of the fund.
Discovery 3 - Yield is (kinda) useless when considering the actual total return of the investment, as $10.3 is clearly better than $6.3 even though the yield is lower in the former case.
Dividend yield is useful for determining what kind of income you can expect from a fund. If I'm trying to decide between two funds (or between two individual stocks) that I want to invest in for income, yield allows me to compare the two even though the share price and the dividend payment amounts are different for each. After I've owned the fund for a while, I can look at the current yield and decide if I am still getting the income that I want from my investment or if I want to switch to something else. It is true that it does not take into account changes in value of the fund. However, if a mutual fund drops from $10 to $6 as in your example (a 40% drop!), you might expect the dividend payments would also take a hit, as it means that the stocks that make up the fund are not doing well at the moment. If the dividend payment does really stay the same as the fund value drops, then yes, you would expect the yield to be higher, and the higher yield would signify to a new investor that they could expect a higher income on their investment.
Discovery 4 - If discovery 3 were true, I then should always look into the YTD, which takes into account the actual share price, but not the yield, of a fund when I try to decide which fund to invest.
Yield only shows us the dividend income of a fund/stock, but does not reveal the total return of an investment. It is only one piece of the puzzle when analyzing an investment, and yes, the year-to-date return is another tool to compare the performance of different investment options.
Your response to discovery 4 is, IMO, an "either or" opinion in that BOTH yield and YTD are just tools and are equally important, not one better than the other. Is my understanding correct? If yes, can you elaborate on the value of YTD v yield from the fund/stock "owner" perspective, not from a "future owner" perspective.
Yes, both are tools to evaluate the current performance of an investment. Both of them are a "snapshot" of the current situation, and don't tell you much about what the fund has done in the past. YTD return tells you how much the fund has increased or decreased in value since January, but because I typically invest for much longer lengths of time than one year, YTD return doesn't tell me much. I'm much more interested in what an investment has returned over the last 5 or 10 years, or longer.
The same tools that are useful to a new investment are also useful to a current investor, because you always have the option of selling your current investment and buying something else.
Some people are investing for income, and the dividend yield is a quick and easy way to determine which investments are providing the best income for a given investment amount.
While "high yield" indicates a possibility of a high income/return for a new investor, at the end of the day, if the share price drops, which will be reflected in YTD but not necessarily in yield, a high or low YTD will be a better barometer for the return of a fund/stock than a high or low yield. Is this correct?
As I said above, YTD return is a relatively short time window, and which tool is better depends on your strategy. When a price drops, it could indicate to you that the fund is performing poorly, but it might instead indicate to you that the fund is selling at a very good price and should be bought, because the price will rise again. It is only one tool among many to evaluate an investment.