The ideas of chirality and spin direction are definitely universal, and not specific to bilaterally symmetric species like us. While we tend to use left/right terminology for both, you could proceed from the other direction, defining "left" and "right" from spin or chiral terms. Once you've established the baseline terminology (in whatever direction), you can apply them even in cases where your original source no longer applies (e.g. tidally locked planets, stars, black holes, galaxies).

We left/right for chirality because our hands are chiral (mirror images of one another, and thus impossible to transform one into the other through any combination of rotation) - I'm not really sure how a species without such common chiral features would use to define chirality - but there are many possibilities (Wikipedia Chirality) - though I'd put my money on clockwise/counter-clockwise rotation. I'm also not sure why our hands are chiral from an evolutionary perspective, or whether it would be likely an alien species would develop something similar.

We use it for spin direction partly because of the chirality, allowing us to simply (and universally) specify whether a spin is "up" or "down" - the actual terms are completely arbitrary. As an example, we use the "right hand rule" to determine the spin of the earth - curl your fingers in the direction of spin (eastward) and the direction your thumb points is "up" - or North. The arbitrary aspects unique to our physiology/terminology are "up/North," "east," and "right." Simply substitute your own terms. Notice that galactic directions will also need some sort of up/down-direction component - rimward/coreward, spinward/trailing are two of the three coordinates necessary to specify an object in three-dimensional space. If you prefer spherical coordinates, you'll need an angle from the galactic plane, and if you prefer cylindrical coordinates, you need a height above/below the plane. Given the general shape of galaxies (round and mostly flat), I'd lean toward cylindrical as the most logical choice.

I'd probably use alternative coordinate systems in-system (cylindrical or spherical centered on the star or system's center of mass) or on-planet (cylindrical, spherical, or surface-mapped rectangular). Another potentially interesting development would be if your baseline or primary species were avian or aquatic - they're probably have brains more specialized in conceptualizing 3d systems - humans are notoriously bad at it, since our brains were developed primarily for flat plains.

Tidally locked planets certainly experience standard day/night cycles and rotating starfields, just like any other planetary body. The primary difference is that their "days" are measured in weeks or months. The primary difference when inhabiting a tidally locked planet is that daytime duration - you have to be able to handle the long-term extremes of heat and cold. Imagine living under noon-day sun for 1.5 months, or the same period of time at midnight - that's what it's like on Mercury (if it were conveniently relocated to Earth's orbit). The most common way to colonize such a planet is to start with the poles (which are conveniently located between day/night extremes), though you could try something more exotic (like a traveling colony - maybe mounted on treads (or legs - like the bad-guy city in the John Carter movie), or perhaps flying/floating?