Hi, I got tagged. I’ll try to help. One thing that is always important is to give the location (even if not exact, knowing the area the photos were taken is helpful). It often makes a huge difference in helping with identifying. Until there is more to go on, a couple of the identifications will remain tentative.
Based on the photos I’m going to guess that you took the photos somewhere on the Colorado Plateaus (perhaps e. Utah or w. Colorado)? If so, there are a number of species present in the region, though several of them do not often appear in plant guides or floras. It is confusing when a book only lists two or three species and there are really perhaps six or eight present in a given area, but this is generally the case. It is often resolved in people’s minds by calling everything “hybrids”. Hybrids do occur, but they are not particularly common.
The first photo is of Opuntia tortispina. The flowers, juicy fruits, and seeds are similar to those of O. phaeacantha, but it is generally a smaller plant with areoles closer together, and usually appears spinier (often not because there are more spines, but because they are closer together). On the Colorado Plateaus, the flower color is highly variable, so isn’t much help in idenfication (the stigma lobes are green though, where they are varied in color, but often pale whitish or yellowish in O. phaeacantha).
The second is difficult to tell, it looks like a plant that grows in shade most of the day. It also looks like there is a broken beer bottle bottom behind the plant, which gives some potential size reference (a largish plant for the region - if I guessed correctly). I would need to see the flowers, fruits, and ideally new growth (with leaves still on) to be certain, but my best guesses would be O. woodsii, O. gilvescens or O. phaeacantha. Based on what I can see here, I think O. woodsii is most likely, but more information could change my mind. Flower color won’t help much here, but the structure of the flower can. As for color, O. woodsii is usually orange, pink, or red, while the other two are usually yellow (O. phaeacantha often with an orange or red center, but gilvescens rarely). O. gilvescens usually has rounder pads than this, and phaeacantha usually somewhat wider pads with more and longer spines. O. woodsii will have mostly largish elongate fruit. O. phaeacantha will have smaller fruits than the other two - simlar to those of O. tortispina (but probably larger than those), roughly barrel-shaped, and O. gilvescens will usually have largish broad fruits. All relative though - they will all have fruit smaller than some of the bigger more southern species. It may not be any of these species though - there are other options, but none seem to fit your photo very well.
The third photo is probably Opuntia polyacantha (though the fruit still present looks abnormal - like it dried out before mature - so it doesn’t help much - it could be a hybrid). It appears to be of a variety of O. polyacantha found on the Colorado Plateaus mostly in Colorado and Utah (leaking into Wyoming and Arizona a tiny bit), and also in desert areas of the northern Great Basin in Utah, Idaho, and Nevada. This plant has been given a few names, the oldest of which is Opuntia rhodantha (another is Opuntia xanthostemma, based on a different flower color). O. [polyacantha var.] rhodantha has never “legally” been made a variety of O. polyacantha, but that’s what it is. In Lyman Benson’s books, this sort was lumped in with the rather similar variety “utahensis” under the “species” O. erinacea. O. erinacea is a VERY different plant, and also belongs with O. polyacantha. The name erinacea has “legally” been made a variety of O. polyacantha, but the other two names didn’t tag along when that was done. So, it’s basically a matter of book-keeping; the names need to basically be filed correctly under O. polyacantha, following the botanical code, be “legal”, even if they are the correct names. So, biologically this looks like “rhodantha”; a variant of the O. polyacantha group that is commonly grown in gardens where winters are cold. AND, I could be totally wrong - it could be O. polyacantha crossed with something else (such as O. tortispina).
Sorry, I tend to be long-winded sometimes.
Hope that helps. I know it’s a little wishy-washy at this point.
If you have more photos, and if you can tell us where they were found, that would help to be a little more certain.