For example, a story today talked about SPY approaching support levels. Given that the price of the ETF is essentially determined by the price of its constituents, does this even make sense? Do people say "oh, the weighted average of all these stocks is hitting a support level, so if it goes below it, then it's likely that all (or most) of the stocks will go down in price"?
Yes, it makes sense. Like Lagerbaer says, the usefulness of technical indicators can not be answered with a simple yes or no. Some people gain something from it, others do not.
Aside from this, applying technical indicators (or any other form of technical analysis - like order flow) to instruments which are composed of other instruments, such as indexes (more accurately, a derivative of it), does make sense.
There are many theories why this is the case, but personally i believe it is a mixture of self fulfilling prophecy, that the instruments the index is composed of (like the stocks in the S&P500) are traded in similar ways as the index (or rather a trade-able derivative of it like ETFs and futures), and the idea that TA just represents human emotion and interaction in trading.
This is a very subjective topic, so take this with a grain of salt, but in contrast to JoeTaxpayer i believe that yields are not necessary in order to use TA successfully. As long as the given instrument is liquid enough, TA can be applied and used to gain an edge.
On the other hand, to answer your second question, not all stocks in an index correlate all the time, and not all of them will move in sync with the index.
With the disclaimer that I am not a technician, I'd answer yes, it does. SPY (for clarification, an ETF that reflects the S&P 500 index) has dividends, and earnings, therefore a P/E and dividend yield. It would follow that the tools technicians use, such as moving averages, support and resistance levels also apply.
Keep in mind, each and every year, one can take the S&P stocks and break them up, into quintiles or deciles based on return and show that not all stock move in unison. You can break up by industry as well which is what the SPDRs aim to do, and observe the movement of those sub-groups.
But, no, not all the stocks will perform the way the index is predicted to.
(Note - If a technician wishes to correct any key points here, you are welcome to add a note, hopefully, my answer was not biased)