what I mean is do singers sight-sing on the first run say on a gig, and sound relatively good, equivalent to sight-reading a piece for musicians? like in a gig, the accompanist has to play sight-reading.
Some singers in some situations absolutely read cold and sing very well.
I did a home recording once with two professional singers and a professional accompanist. We sent the music out about two days ahead of time but none of the three needed to rehearse it at home. We made about three takes of six different songs. First take was ok and they all got through the numbers but we gave them performance notes. Second take was often spot on or at least very good - including acting out the emotions of the scene. Third take was either a safety or alternate read or maybe a fix of one or two moments.
Professionals do some amazing stuff.
For professional studio gigs, sight reading, including for singers, is a must. Studio time is expensive, so musicians are expected to be able to handle their parts very efficiently.
For performance gigs, singers don't generally use sheet music, so learn the music ahead of time. The exception would be a last-minute emergency. I played in a blues band that needed a singer at the last minute, so we brought in someone who used lyrics sheets with notes about form, but not notation. On the other hand, in a last-minute substitute performer in an opera would have already learned the role.
Singers absolutely improvise. In jazz, of course, scat singing is a well known form. But there are other kinds of improvisation as well. On one gig, we'd written the song minutes before the gig and hadn't had time for lyrics. The singer did a sort of "mumbling" so that it sounded like he was singing words but was actually singing nonsense. Somewhat similarly, I was in a musical and forget words to my big solo -- I had to make up lyrics for a complete verse.
Another place where one can encounter singers improvising is in comedy sketches. For example, the audience might be asked for a theme, and then a song is improvised based on that theme. Or perhaps a well-known song is sung in a style given by the audience.
From my experience, it's the other way round. A new vocalist will turn up with their orchs, and expect the band to sight-read them, while they sing the stuff familiar to them. There have been gigs where a new vocalist will simply sit in, and sight-read the existing charts, but it's nowhere near as common. But good ones are more than capable, cold. And a pleasure to work with. The ever present problem is the range of the songs. Some vocalists have enough range to cope with most, whereas others would not be physically able to sing in the keys the charts were written in. And not many bands would be able to transpose at sight to accommodate a singer with a limited range. That's a tall order!