What is the vertical axis of this waveform shown in this tweeted IRIS GIF?
Answer 1

The horizontal axis is indeed time (don't forget units and note how long those periods are!). As for the vertical axis, I'm certain that this is displaying vertical (or 'Z' or 'V') displacement. Keep in mind - like you mention - that we are only looking at a 1-D picure of the data, but there are other perspectives we are missing! For example, how is the earth being displaced with respect to North-South and/or East/West directions? 

Answer 2

The vertical blue line you reference indicates different seismic phases (or arrivals). For example, you list 'P' and 'PcP' - but there are many others! For example 'SKS', 'SKKS', etc. - see the hyperlink to understand this notation and the seismic waveforms they correspond to.



NOTE(1): For the term phase - be VERY careful with this word and the context I've applied it in. As is customary in geophysics, one word can refer to the same or different concepts. Read here for a good follow-up example of what I'm describing in terms of phase. 

NOTE(2): Many seismograms like this will initially be displayed in the conventional NEV (or NEZ) or North-East-Vertical coordinate system . But, to optimize many kinds of analyses and displays, one needs to rotate (or transform) this system into another - LQT. Read here to understand what that means. 

NOTE(3): How does one know or estimate seismic phases like these? You need an earth model! From the source location and at the instantaneous moment of when the seismic event occurs, a corresponding seismic wavefield is generated and travels throughout the earth - interacting with its fluid and/or elastic mediums - all the while responding to (or convolving) its complex, 3-D velocity (and density) structure. As the seismic wavefield passes through different mediums via refraction, reflection, dispersion, 
attenuation, etc., it changes and subsequent various phases are produced and then recorded at a station. The phases - i.g. 'SKS', 'P', and many others - basically tell a story of how/where the wavefield changed, which is great for analyzing the crust, all sections of the mantle, and even the core!

*I've included a lot here but you've asked an outstanding question that you (and I) can really learn from!