For example, when I use ND1000 filter, I know theoretically I have to increase shutter speed by 10 stops. Does this apply regardless of different stop increment settings (full stop vs half stop vs 1/3-stop), or do they say 10 stop in full stop increment?
It sounds like you are confusing "stops" with "steps". A photographic stop is a doubling of light exposure.
If you can adjust your exposure by 1/2- or 1/3- stop increments, then you would need 2 or 3 increments adjustment (respectively) to achieve a 1 stop change in exposure. So when using a 10-stop ND filter, you would need 20 steps if you were adjusting in 1/2-stop increments, or 30 steps if using 1/3-stop increments.
If you were shooting a scene, without ND filter, and your exposure settings dictated a 2-second shutter speed, then a 10-stop ND filter would necessitate increasing the exposure by a factor of 210 (i.e., 1024). Thus, you'd set the shutter speed to 2 * 1024 = 2048 = 34.133... min = 34:08.
Note: a ND1024 filter is precisely a 10-stop (i.e., 210 = 1024 multiplier) ND filter. A ND1000 is actually a multiple of 1000, not 1024. Thus the change from 2-second non-ND exposure would be 2 * 1000 = 33.33... min = 33:20. However, the ratio difference of 1024/1000 is only 2.4%. In terms of stops, the difference is log2(1.024) = 0.034 stops, a trivially small exposure difference that you won't be able to discern between two images, and way below the smallest precision of any commercially-available DSLR or MILC camera.
Regarding fractional-stop ND filters, they aren't common, but they do exist. I use mostly Lee filters. While I don't own fractional-stop filters, Lee makes ND 0.45 (1.5 stop) and ND 0.75 (2.5 stop) filters. If one were to use either of those filters, the exposure settings would need to be adjusted by 21.5 ≈ 2.83, or 22.5 ≈ 5.66, respectively.
Now, regarding fractional-stop exposure settings in camera: it doesn't matter if your camera is able to set exposure settings in fractional-stop increments. Whatever the "correct"/desired exposure settings a particular scene calls for without a ND filter in front of the lens, you need to increase the total exposure by the number of stops as called for when using your filter.
For instance, let's say a particular ISO 100, ƒ/8 shot is metered for 1/50" . A 10-stop ND filter would still require increasing the shutter duration from 1/50" to 1/50 * 1024 = 20.48 seconds. I'm not aware of any cameras or shutter remotes that allow fractional seconds beyond 2.5", so you can just pick 20 or 21 seconds. The difference is only 0.07 stops (i.e., less than 1/10 stop, still less than any camera can control for).
Related questions and links:
- Are neutral density ratings in 1/3 stops or are they really in 0.3 stops?
- How to read ND filter description?
- Choosing between 9-stop and 10-stop ND filter for flowing water long exposures — While you aren't asking the same question, the answers address similar concerns where a small difference in stop calculations don't matter as much with 10+ stop ND filters as it does with 1–3 stop (or so) ND filters.
- Neutral-density filter article at Wikipedia. The "ND filter ratings" section in particular is useful regarding the different nomenclature for ND filter strengths used by different filter manufacturers.