Will I have problems installing multiple desktop environments?
I currently have Unity and Gnome 3.6 installed on my computer and I want to check out the KDE desktop environment. Is there any problem with installing multiple desktop environments on my desktop?
You won't have any problems installing multiple desktop environments. Nothing in the system will stop you from installing KDE, GNOME, Unity, Enlightenment, and all the others all at once (except for disk space).
However, desktop environments will often "argue" with each other and overwrite settings. For example, installing KDE on a system will very often break a Unity installation by overwriting GTK or similar properties. Similarly, installing Unity will break KDE most of the time.
Unfortunately for you, these incompatibilities are very hard to repair after they've happened, so this is a risk you must be willing to take with your system. If you want to explore different environments, I'd highly recommend you use VMs until such time as you find one you like. Then, install that one and only that one on your system.
That is rather difficult to answer depending on what will be considered a problem and on your level of tolerance to what are arguably small or negligible problems. (Some people are very tolerant to errors in Linux just because they haven't payed for it.)
You can install as many desktop environments you want, but... is that reasonable? - Isn't that even crazy? - If the camel's back didn't brake it doesn't mean we have to try to break it.
There are also problems and bugs that can happen, and conflicts are also possible even without multiple desktops.
I think these risks will increase the more DEs are installed on the same system, the greater the difference is between them, and the heavier/complex they are.
The greatest difference I can see is that between KDE/Qt and GTK systems. Unity, Gnome and KDE/Plasma are all rather large and complex (compared to Xfce. LXDE, Openbox, Fluxbox).
As for my own experience, I have always had problems with multiple desktops, usually only in one of them. That is: there may be a big chance to have some problems at least in one of them.
I wouldn't dream installing as many desktops environments as possible, whatever that means.
Want KDE? Go for Kubuntu.
Want Xfce? Go for Xubuntu.
Want LXDE? Go for Lubuntu.
Want GNOME? Go for Ubuntu GNOME.
Want GNOME alongside Unity alternatively?
That seems reasonable, but some say no on that also: https://askubuntu.com/a/491049/47206.
Just to test them on the short term, one could go with a virtual environmen (VirtualBox), a bootable live-usb system, or even a separate partition in multi-boot. The latter is more complicated (you have to know what you do) but I prefer it to mixing DEs.