As you have found out, there is at least one scenario where the sequence order in which the data is projected matters, and that is when you mosaic (aka merge) raster tiles. In that situation you must mosaic before projecting. Otherwise, gaps appear between the mosaicked tiles.
I'm not aware of any other sequence-dependent projection issues, but my experience may be limited; others may have additional information.
However, I'm particularly interested in the second question of your post's title, "Or does it even matter?" I suppose that depends on what you mean by "it", so I'll rephrase the question as "Or does establishing a project Coordinate Reference System (CRS), with associated projections and transformations, matter?"
To that question: If your project requires any analysis (buffers, intersects, spatial joins...) then you must establish a single, appropriate, project-based CRS up-front, and project any data that will be subjected to analysis to that CRS a priori.
Why? Because analysis tools rely on the internal location values stored in each layer when doing their computations. The tools will happily do their job, even with layers containing wildly incompatible values (say, merging one layer with UTM meters to another layer with Lambert Conformal feet). In such cases the output is likely to be non-sensical.
GIS staff are often confused when this analysis problem occurs because modern GIS software will display layers with divergent CRSs so that they line up, using "on-the-fly" (OTF) projection, and users thereby assume that the results of data analysis will also magically line up. Not so! Early GIS software did not have OTF display projection capability. Thus, when two layers with different CRSs were displayed on-screen, you could clearly see that they did not line up, and intuitively knew that any analysis with those layers would result in problematic output.
The arrival of powerful desktop computers allowed for OTF display of data with differing CRSs. Although useful for display and map production, OTF no longer provides the user with visual clues about the underlying data, lulling them into thinking that analysis will likewise line up, resulting in a huge source of GIS/mapping confusion.