Be careful of latching relays, the state they come up in is random. (OK, it's determined by whatever happened to things mechanically -- so, random.) This can be bad. I tend to avoid them for that reason, but sometimes you have to deal with it.
The minimum current is required to keep the contacts clean. It applies in the CLOSED or closing condition so of course voltage across a dead short will be small. This can get kind of weird so I'd suggest you search it (relevant term: wetting current) rather than me trying to explain from a phone.
If you are worried about low level signals you want fancy contacts. Pd is said to be the best. I don't know about best but I've used a lot of Pd or Ru contact relays (Panasonic AGQ, Fujitsu FCL FTR-B3, TE Axicom IM) and they seem to do very well. 3V coils and below are widely offered but I don't know about stocked. Somehow, manufacturers actually managed to standardize these things, so they're interchangeable! For once.
Since you want to handle 250V, I would look at the TE Axicom IM series. It looks like it would do everything you want, if there's stock of an acceptable part number.
Otherwise, in light of 250V being a very rare rating for a signal relay, consider whether (1) you need to switch it at all (no switching means you're on the much more forgiving standoff rating table) and (2) whether you can bring in another series switch (maybe semiconductor?) to take the voltage stress off the relay.