It will always be difficult/expensive/impossible to integrate mainframe and distributed products, including centralised control capabilities, when the products come from different vendors.
The answers to the remaining questions are all: it depends on what the customer needs, budgets and so on. It is technically possible to run both mainframe and distributed workloads, from either (for example) CA7 or AE, without needing both. These CA products can use the same System Agent, so if the requirements are to run lots of distributed with a few mainframe jobs, you could deploy AE/DE and also include scheduling to the System Agent for z/OS.
DE v AE - again this depends on the customer requirements. These products use the same Agent and offer almost all the same job types, but the back end server engines work in different ways.