I use water rather than oil - both on the bottom and on the sides. Oil is more durable, so you may not need to re-oil as you work, but if you carry oil in your bag, it can spill or leak and make a real mess, as I learned from experience. Perhaps you could use an oiled cloth - I've not tried it. An oiled cloth is also a good way to coat the blades for rust proofing, especially if you are not going to use them for a few days.
Also, for reasons beyond my knowledge, oil is generally considered to be problematical on diamond dust stones, like one the "coarse" Pro-Filer stone. (The "fine" Pro-Filer stone does not use diamond dust, but I still use water.) (AFAICT, most modern sharpening "stones" aren't really natural stones, but are an abrasive powder embedded in an adhesive.)
Of course, water wouldn't work very well at machine sharpening speeds (would probably boil away), but is often used in a variety of hand sharpening situations. E.g., some fine woodworkers use it to sharpen their tools. Water and oil don't mix well, and if you have got oil into your stone, you may want to wash it out thoroughly with water.
Generally, I only use tape if the tool gap is too wide, making it too hard to keep the blade centered on the "stone". (I don't have any side honed blades that need tape to compensate.) Water is a good enough lubricant to avoid scratches.
You lubricate the sides to prevent scratches, which can at least theoretically slow down the blade by creating friction. You lubricate the bottom for the same reason, and also to create a cleaner cut, just like you might lubricate a metal working machine tool.
If you use tape:
Are you putting the tape on the blades or the tool? If on the tool, you should figure out a good way to put pressure on the tape, so it sticks well. E.g., put a little extra tape on the blade, so it is a very tight squeeze, and leave it there a minute. BTW some tapes, like plastic electrical tape, are made with adhesives that are only fully activated when you stretch the tape.
It never occurred to me to leave tape on the blades long term. If that is what you are doing, and you worry about appearance, I guess you could use a tape which doesn't show too well like 1/8" or 1/4" wide aluminum foil tape. A gold foil tape might even look pretty, in which case it could be wider.
Or maybe some other pretty colored metallic tape. Sort of like a custom colored blade, but done by you. You could even make one side of your blade one color, and the other side different, so it flashes alternate colors as you turn. But that draws attention to the blade, so be neat.
I no longer use tape to center the blade, to compensate for any asymmetry in the tool. Instead, I take a few strokes in one direction, turn around the tool or the skate, and take a few more strokes, etc. That evens things out.
What ROH do you use? If it happens to be 1/2", you can use one of the old
Berghman Skate Sharpeners which were much better designed than Pro-Filer, because you can adjust the gap. (Though I admit the Pro-Filer looks a lot prettier.) Then you can leave the tape on the tool very easily, because you have better access to the bare tool, and can push the tape on more strongly, and use cloth athletic tape or even duct tape, which doesn't come off so easily. Also, you can place the stone right at one end of the holder, which makes it easier to control and avoid hitting your toe pick, though I would still tape the toe pick to protect it. The old Berghman stones were fairly crumbly, especially if you get a 1930's era tool. And they were very coarse, which is great if you want to work fast, but may not create as nice an edge. Since you already have a Pro-Filer you can take the stone out of the Pro-Filer, and put it at one end of the Berghman (the Berghman was designed for a longer stone).
If you don't happen to use 1/2" ROH, you are stuck with the Pro-Filer. The only other commercially-made high quality skate hollow cutting tools I know of are designed for hockey blades, which are much thinner.
Anyway, I hope you recorded your rocker profile, by photocopying or tracing the blade. Even hand sharpening will eventually alter the profile, if you aren't careful. Also, be sure to re-emphasize the sweet spot(s) (point(s) up where the rocker radius changes.) once in a while, so you can continue to feel it. It is possible that on a Coronation Ace blade, there are two sweet spots up front, separating 3 rocker radii.
The pro shop might not appreciate you showing other people at the rink that you can sharpen blades. Sharpening is most of how they make a living.
On the other hand, it is another service you can offer your students - quietly.
A micrometer or precision calipers isn't all that useful on a parallel-side ground blade. Your blade either fits the tool gap width, or it doesn't. It doesn't matter why. However, if you are curious, it lets you measure how much metal you are taking off with each sharpening - though, you probably won't take off much, especially when using the fine grain stone. If you sharpen reasonably often, a minute or so of strokes with the fine grain stone may be all you need. In fact, a lot of the time, you aren't actually trying to make a new edge, but are just straightening and polishing and cleaning the old one - the other guys posting here have talked about using a steel instead of an abrasive stone to do that, and remove even less steel.
Anyway, I hope you enjoy playing with your new tool, and that it works as well for you as when you had an expert to work on your blades.