I am asking for a single device that can not only charge all common battery types, LiPo, LiHv, LiIon, LiFe, NiMh and NiCd. But also it can test them for capacity, health and faults automatically. Basically, you put any (rechargeable) battery insode and the device tells you if the battery is ok to use, what current and max capacity are, all while the battery gets charged.
If such a device does not exist, what would come closest?
Chargers for RC model vehicles (car, airplane, helicopter) can do most if not all of that stuff, but you will have to be comfortable with soldering connectors as there is no universal standard connector system for any of the battery types you mentioned, and even standard size 18650/21700 cells are rarely used for RC purposes. The RC hobby has mainly settled on XT60 and its smaller cousin XT30 as the closest you’ll find to a standard, but even within the hobby many batteries use other connectors. Snipping leads and soldering connectors is not an optional skill, the currents involved can be very large and will easily melt a poor connection made with poor skills or some hacky clip-on connector.
For charging, this is the sort of thing I use, no promises. RC chargers also include a balancing system to allow it to balance different cells across an entire battery pack but you will have to have individual wires junctioned in between each of those cells so it can sense their voltage and top them up as needed.
Also most RC chargers don’t bother having anything to do with lead acid (automotive style 12V or otherwise), as they are much too high amperage and heavy for any sort of RC use and they use a wildly different charging design and have much more complex health monitoring and maintenance needs. Not recommended for that, use an automotive, marine or off-grid style battery maintainer and repairer for those. The one I linked says it does handle SLA (sealed lead acid) but I wouldn’t trust it to do a good job. Compromises have likely been made. You’d be better off with a dedicated unit for those if you are going to be dealing with them.