Are religion and morality separable

Answer 1

Certainly not.

Humanism is a collective term for a number of philosophies dating back to ancient Greece, India and China; what they have in common is an emphasis on human concerns without reference to the supernatural.
You could say that they derive a system of morality by reference to humans rather than by reference to gods.

Morality in practice need not differ all that much between religious and non-religious systems either. I suppose pretty much everyone thinks beating up old ladies is a bad thing to do, whether they are religious or not.
Disagreements between religious and non-religious moralities tend to be in areas where it is hard to see that the behavior will hurt people, where a religious morality can still refer to the opinion of gods.

The relatively modern concept of human rights is an example of a moral view I think a large number of people can agree on, more or less independent of religious views.

For more on humanism, see related link.

Answer 2

Yes, you can have religion and no morals (pedophile priests), and morality without religion--many people have ethics but no particular religion.

I'm pretty sure most of the people on death row call themselves religious.

Answer 3

Ethics continually change, often drastically. Social mores come and go, with certain practices deemed acceptable in one era or area, but capitally liable elsewhere. Each new civilization seems to have different ideas as to what is acceptable.
Therefore, a higher ethical standard, not invented by humans, can serve as a permanent criterion and a safeguard against excesses.See also:

The value of religion