Personal choice would be CentOS 5.3. I haven't had a problem installing it on anything yet. As for developing software, during the install, just make sure to install all the development packages that you will need (there are a lot of choices for language, editors, etc.). Even if you don't install them during the install process, you can always use the package management tools to install what you need later. And, apache, mysql and qt are available. But, I am pretty sure that just about every distro has those packages available. If they aren't installed in the base install, they are available as packages or can be compiled from source.

As for support... If you mean Redhat Enterprise linux, then you do get support from Redhat, but you can't install enterprise without paying for it (as far as I know). CentOS (basically Redhat enterprise stripped of proprietary stuff) and Ubuntu are free, and the support is the multiple forums all over the internet. I've been developing in linux for many years, and there's usually someone out there that knows the answer and is willing to help.

Yeah, basically if this is a commercial/serious enterprise, go with RHEL 5.3 and get the support ctc with RedHat until you are comfortable. Your management may even insist on it anyway.
If this is just for personal use, go with the free equiv: Centos 5.3.

Just FYI, you could get hold of a copy of RHEL without a support ctc eg free trial version, but you won't get support from RH and you won't get any updates .. not recommended except as a trial run.