I don’t think I’ve ever measured the final string gap at the first fret as I do it all by eye, but basically as low as you can get without the string buzzing on the first fret. I did try and use a nut file as a gauge/cut stop on my last set-up but ended up with the string flat on the fret, so replaced the nut and went back to doing it by eye. I don’t really like it when a specific height is mentioned, as a lot depends on just how much neck bow is set and just how level the frets are (my fret levelling improves as I do more of them, but I occasionally have to re-do the level when I can’t get the low action I know I should be getting). You can often achieve a much smaller gap. However, the smaller the gap, the more risk there is of a first fret buzz, so if going for the ultimate low action, you may go too far, so you need to be prepared to either fit another nut, or do a slot-filling repair job with super glue/CA and baking soda/bicarbonate of soda.

The thicker the string, the bigger the gap needs to be, but with level frets you should be able to get away with the same distance above the first fret as at the second fret when fretting at the first. You can always add a little bit extra to be on the safe side.

Lowering the nut slot height will bring the overall action down, but unless the nut is very tall to start with, in terms of the risk of buzzing, it’s only that first fret when playing open strings that you need worry about. If the nut slots are very tall, then there can be a greater vertical element to the string tension on the neck (adding some bow), which means that when the nut slot heights are lowered, the neck curve may flatten slightly, so you may need to slacken off the truss rod by a very small amount to get back to the curvature you set up before working on the nut.

But it is the work of moments to adjust, and I generally still play around with the truss rod setting and the bridge height after I’ve cut the nut slots to see if I can get the neck a little bit flatter. But then I do like really low actions, so if you prefer a reasonable gap under the strings so you can really dig in, then you may want to leave it well alone. But even with an action that’s generally on the high side, you want to have strings that are easy to fret at the 1st fret. Also, too high here and the strings will be sharp when fretted.

If you’ve set the intonation up before lowering the fret slots, then you’ll need to check that again. As previously mentioned, lowering the slots will bring the overall action down, which means you need less force to fret a string at the 12th fret. This means less change in string tension between the unfretted and fretted condition, so the 12th fret note won’t be as sharp, so the saddle will probably need to come forwards a bit, especially when you have taken the nut height down significantly.