How can they initiate contact with such a fast lived race, and how can they engineer events such that a race with lifespans and social structures similar to humanity can remember them and (perhaps most importantly) respond in order to engage in some form of conversation?
The same way you already communicate with hyper fast entities to whom human language is unintelligible: with a compiler!
What's a compiler? It's a computer program for translating thoughts expressed in human language into machine language.
Bear with me here - you often instruct your computer to do things on your behalf. Compared to computers, you and I think at a positively glacial pace; let's say you subvocalize at a rate of approximately 300 words per minute. Thus, the upper bound at which you personally can process, say, a tweet (140 characters = 28 words) would be
$$\dfrac{\frac{300 \text{ words }}{\text{min}}}{\frac{28 \text{ words }}{ \text{tweet}}} \approx \dfrac{10 \text{ tweets}}{\text{min}}$$ (or 6 seconds per tweet).
So you can process tweets at a rate of $\frac{1}{6}\mathit{hz}$. By comparison, a single CPU computer can run machine-level instructions at a rate of $4\mathit{ghz} = \text{4,000,000,000}\mathit{hz}$! Even if you assume it takes 100,000 instructions for a computer to process a single tweet, a single CPU is still processing tweets 250,000 times faster than you can. By analogy, an entity that processes a tweet 250,000 times slower than you would take 17.36 days to process a single tweet.
So in the same way that people have been able to create compilers that transform thoughts expressed in human languages at human speeds into machine instructions operating at computer speeds without ever having gained the ability to directly work at computer speeds, these ancient and wizened creatures will need to specify their messages at their own plodding speed and then compile it down to a message that is broadcast at the much higher experiential speed of the youngling species.
Because any given individual observing these messages won't live long enough to see the Old Ones compose a reply, the "Hello Universe" message the Old Ones choose needs to convey a lot of information up front. 634
At a minimum, it needs to:
* Indicate to whichever young species may be listening that it is a signal devised and sent by another intelligence.
* Indicate the time scale on which a reply from the Old Ones should be anticipated. Failing to communicate how slow this conversation will be will likely lead the speedier races to assume the artifact they picked up the broadcast from was a relic of a disappeared species. You want some sort of galactic timescale hourglass here.
* Make evident how the recipient can respond. If there is a physical artifact broadcasting the message, maybe it just has a switch that can be flipped by the young whippersnappers which will indicate "somebody found the thing and flipped the switch to say hi". It should be evident on the timescale of both species that the thing is calling home to report contact. 350