Will a small "breather tube" on my bottles harm the oil/vinegar/liquor inside?
The exposure to oxygen that you'll get with bottles like these will affect your ingredients, but that will vary depending on how long you're storing them.  If you go through enough oil, vinegar, etc. that you're refilling these every few weeks, I think you'd be just fine.  If storing for three months or more, you may start noticing your oil getting slightly rancid and your vinegar getting slightly oxidized, or you may not start noticing the difference until much later.  A lot of other factors will impact how quickly this progresses - exposure to light and a higher ambient temperature will result in faster quality declines.  If you're not refilling these any more often than every six months, any gains in efficiency likely won't be worth the impact on quality.

There's some evidence (see here) that leaving ethanol solutions (like liquours) exposed to air can result in evaporation and a change in the ABV, but I would think that exposure from a 1mm hole would be negligible and you could keep these bottled for multiple months without much impact on quality.  Again, if you're going through such limited quantity, your efficiency gains will be pretty minimal as compared to keeping your liquors in their original bottles.  I wouldn't recommend storing or pouring liqueurs from spouts like these; in my experience, the residual sugar can build up over time and restrict flow through the pouring spout.  Sugary liqueurs will also attract fruit flies, and it's possible that a particularly small and persistent bug could find its way down the air hole.  You can use plastic wrap or buy plastic covers to place over the top of your spout to help ward off flies - any reputable bar will cap their pour spouts overnight.

If you keep vermouth or other fortified wines in these, store them in the fridge.  Dry vermouth in particular will oxidize quickly and develop off flavors if exposed to air, like any white wine - lowering the temperature will help somewhat.  Sweet vermouths and sherries are a bit more stable but not indefinitely so, and their higher sugar content can lead to buildup like liqueurs.

If your ultimate goal is consistency, be warned that professional bartenders typically practice extensively before attempting to "free-pour" and hit accurate amounts with speed-pouring spouts like this.  The reason is because differences in ABV and viscosity cause differences in pouring speed, so you'll wind up with different volumes depending on what you're pouring.  Even if you're very accurate with your pouring times, you'll need to train yourself to deal with these differences.  The best way to be consistent in your home bar is to skip free-pouring and measure using a jigger or other device.