Can I download latest .deb file of kernel from http://kernel.ubuntu.com/~kernel-ppa/mainline/ (for example v3.7-rc1-quantal) and install it on Ubuntu 12.04.1 LTS (precise) or any old Ubuntu like 10.04 LTS?
Yes. You should note that rc can be unstable. If you are adventurous and want to customize the kernel, downloading release package(s) directly from http://kernel.org/, building, and installing can be an option. Have fun.
Update: If you have installed virtualbox, vmware, and/or nvidia, you may want to hold off on using kernel v3.7-rc*. These require patches for building kernel modules. Issues are related to VM_RESERVED macro and include/linux/version.h header file. Am sure these issues will get resolved in coming days.