Yes.

You can take binary or trinary star systems and swap one of the stars for a black hole and nothing changes in the orbital dynamics.

Depending on the layout of the solar system planets can orbit the stars, the black hole, or some mixture of the above.

Some of those planets could be in the habitable zone (liquid water).

And some of those planets could develop intelligent life.

We don't know what the probability of most of thees steps is (especially the last one) but we know they are all plausible.

Maybe.

The only mechanisms we know that can form a stellar mass black hole (a few to a few tens of times the mass of our sun) are neutron star mergers (seemingly very rare) and supernova explosions. Either of these are extremely energetic events, and since planets apparently form as part of stellar formation, all the planets in the system will exist at the time of the supernova that created the black hole (neutron stars also form in supernovae, but the time for two to decay orbits and merge would probably exceed the viable lifetime of a habitable planet).

A supernova within a few light years, never mind in the same gravity bound star system, would likely sterilize any existing planets; one within a fraction of a light year (say, even a 500 year orbit) would probably strip the atmosphere from a rocky planet, almost certainly ruining it for future re-evolution of life. Such an event would very certainly destroy any existing civilization that couldn't flee well ahead of time.

You'd need a situation where a planet manages to at least retain an atmosphere (and liquid water or other suitable liquid solvent, say liquid hydrocarbons as on Titan) for life to have a chance to re-evolve, or even for a planet to be suitable for recolonization after the black hole forms. No, not impossible -- the universe is a big place -- but pretty unlikely.

The situation might evolve in a globular cluster where stars ar fairly close together. IF a black hole in moving through the cluster at the right velocity, it might attract stars into an orbital arrangement with the black hole in the centre, and the planetary systems of the stars remaining in orbit around them.

This is going to be a delicate arrangement, since the act of gravitationally attracting the star to the black hole is likely to cause disruptions in the orbits of the planets, kicking planets out of habitable zones, or alternatively moving them into habitable zones.

The likely look of such a system from far enough away will be similar to a comet, with the "tail" being billions of comets and Oort cloud objects from the various star systems, followed by some ejected planets, then the stars and finally the black hole itself, likely visible by the accretion disc made up of the relatively denser dust and interstellar gas from the globular cluster.

Imagine the bright nucleus is the accretion disc of the black hole, and the "tail" is the objects scattered and pulled along by the black hole. The star systems with planets would be about 1/3 of the way back from the nucleus, any closer and they would be roasted by the energy of the accretion disc

The sky from these planets will be interesting. On a planet you still have your sun and other planets (likely in rather eccentric orbits), with several other stars nearby and an amazingly bright object high above the plane of the ecliptic. The constellations will change rapidly (say over the lifetime of a civilization), constellations visible in the early Roman Empire will be long gone by the collapse of the Empire because of the orbit of the stars around the black hole.

If mankind created an artificial black hole that eats earth, then it would have the mass of earth. It would be in the same orbit as earth was and there shouldn't be any big change.

So maybe if a terrible accident happened to a colonized planet or if the greatest mass murder of all history happened during an interplanetary war, then I guess it could.

It is not only possible, it is normal: Earth is a planet orbiting a star which is orbiting the black hole at the center of our galaxy.

A black hole, the likely size to act as a central mass, with orbiting suns, would likely be the result of a supernovae, which would have a typical lower size of about 25 solar masses. The nearby few suns would likely have great orbital distances and/or high orbital velocities. Eventually the black hole will increase in gravitational draw, and then slowly the suns would be sucked in. In other words, this scenario is an unstable one. Additionally, it would involve extremely large orbital distances and velocities. Planets orbiting would have to have high orbital velocities, and like the suns might be subject to substantial tidal forces. Higher orbital velocities incur a higher probability of matter accumulation, which is incompatible with the stability to support life.

So not very likely.

Addendum: Stated differently, the probabilities for "solar systems" of multiple stars, referencing the original poster term of "a few suns" orbiting around a black hole, carries the implication of a larger black hole, and since the observed black holes are in the 10 to 100 solar mass region, with 25 being near the modal for the distribution (and perhaps for other reasons as well), the larger solar mass black holes would permit "orbiting around" of a few suns. Secondly, in the Milky Way it is estimated that most stars, perhaps 85 percent, are red dwarfs, which are generally about 0.2 solar mass. Additionally, estimates (some are less) of the red dwarfs have stellar companions. So to have a black hole, with an orbiting set of a few suns orbiting it, appears to be a rather improbable event, given our contemporary understanding of stellar demographics. However, things do not end there.

To have a system of planets orbiting the back hole with a few suns orbiting it stellar collection, would likely require large orbital distances and/or high orbital speeds, as the central mass would be rather high. That is, not the tiniest black hole, and then again a few suns in orbit. So we are talking a large footprint for the resulting solar system.

The original poster asks if civilizations, which I suppose means "life as we know it" to be inhabiting one or more of those planets. Now the collection of those few suns and the black hole must be distributed in a rather uniform manner to permit somewhat uniform insolation of the possible planet in the green zone. The probabilities keep dwindling.

So we have a mid-sized black hole, with a few suns, somewhat uniformly orbiting it, and those suns should not be of the ever popular red dwarf type, and somehow this arrangement will provide somewhat uniform insolation to a planet sitting in the green zone of the system.

COULD something like this exist? Who could say definitely yes or no? However the majority of solar systems are binary, so that tips the scale towards systems of more than one central star. However, the effects of multiple solar mass bodies tends to have other effects which enter into consideration.

Accretion rates, are likely to cause a doubling of mass from tidally disrupted stars (2x10^10 years approx), and for gravitational diffusion (2x10^9 years approx). There is much more to factor in, but the doubling of the black hole mass within the lifetime of a planet such as earth is real, and would have disastrous effects. For example, the luminous flux of the resulting suns would be compromised, changing drastically the green zone. Additionally the ever cycling tidal forces of the multiple stars would likely create a substantial fluidity of the planet structures. This might also affect the development of civilization.

A few suns in addition to having the right irradiance, would also have to have rather regular orbits, as eccentricity would adversely impact consistent planetary insolation.

So summarizing this, a central mass black hole, orbited by a few suns, with sufficient and uniform insolance on a planet in a green zone, which would be presumably free from excessive tidal forces, and having that central mass black hole be stable and not substantially accumulating mass from accretion or other effects in what promises to be a region ripe with tidal forces and gravitational diffusion, seems rather unlikely.