How does the influence of clouds on climate vary with height?
There are a few things that come into play when you talk about the radiative forcing of clouds, or their effect on the climate budget. "Cloud" is a pretty wide-sweeping definition. Scientists are still trying to fully understand and quantify these effects. Clouds do not vary only in where they are in the atmosphere but also in their phase (ice versus mixed phase versus water) and their depth--the combination of these two affect the cloud's optical depth. All of these different properties affect the net effect (whether positive, negative, or neutral) of clouds on the climate system.

High clouds are generally ice crystals, yes, because they are in areas of the atmosphere that are far colder than the freezing level. They are also generally geometrically thin. These high clouds have little effect on the shortwave energy and instead mainly reduce the amount of longwave radiation leaving the planet.

Clouds in the middle parts of the atmosphere are usually tall cumulus clouds that are full of liquid and mixed phase precipitation. These will be very bright to shortwave radiation and thus reflect much of that incoming radiation. They also provide some trapping of longwave emission from the surface and radiate less longwave radiation for a given level in the atmosphere than would normally be the case. The anvil regions of these clouds, on the other hand, are full of ice crystals which again don't reflect much shortwave radiation but do reduce the amount of longwave radiation leaving the planet.

Compared with the high atmosphere, completely ice phase clouds near the surface are thicker and their shortwave reflection is therefore a bit stronger than their thinner counterparts in the middle and upper atmosphere; they also trap longwave radiation from the surface. Mixed and liquid phase clouds' effects have similar effects as ice phase but are stronger in both the short- and longwave spectra.

Another thing special about clouds near the surface is because of that proximity they can actually have a strong influence on the surface which can cause a climate feedback. If you have a deck of clouds overlying a bright surface, it can actually warm the surface a bit since it is reflecting less shortwave than the highly reflective underlying surface would otherwise. For example, it has been hypothesized that the Greenland Ice Sheet melting is enhanced when cloud cover is present.

A good resource on this topic is available here. There have been many hypotheses over the years as to how these clouds and their effects will change in a changing climate. You can search for things like the "Iris hypothesis"--and subsequent refutation--as a good example of this still-evolving science. Climate models are still struggling to cope with correctly representing all of the feedbacks associated with clouds, which causes such high uncertainty in cloud effects in the IPCC reports.

One last thing to keep in mind is that these forcings are relative to a column with no cloud in the current climate. But what if our distributions or regimes of clouds greatly shift? If the planet warms so much that ice clouds can never exist, the relative forcing of ice clouds in that climate would then be null. So akeep in mind the point of reference you are using when thinking about the forcing of clouds.