Is there any battery-less Android-based device I'be been working on some Chinese (bit crappy) device, with own system, that had one wonderful (at least for me) feature. When constantly powered-up (plugged to power adapter) it could work without the battery inside.

Is there any Android-based phone or device that does the same? Or the only option here is to keep battery inside, keep it powered up all the time (I have some reasons for that) and replace battery quite often, since it will be charging and discharging it the same time (will it), thus lowering its life-span.
Don't you worry. LiIo batteries are not that concerned about that. What really lowers their life span are "complete discharges", not frequent "trickle-charging". It's even recommended to charge them as soon/often as possible (and practicable, of course).

See e.g. Wikipedia: Lithium Ion batteries for details, and also take a look at our question Does constant charging harm my Android cellphone? and its answers.