should I mow?

[tarheelchick]

Hi All! I'm back with a couple of questions...still consider myself as a newbie since last year was my first garden ever.

I've attached a picture of my garden in its current state. Last year when I finished harvesting, I planted rye grass for my winter cover. I literally haven't touched it since then. As I understand it, I should now turn this grass over into the soil and then I'll be setting out my summer plants around mid May. Because of how tall this grass is, should I mow it first? then turn it over? Will it be okay if I till it under with my tiller or do I have to manually fork it under? (that seems like it will be really hard to do). And lastly, I started my first ever compost pile last year, and I now have some compost that I think is ready to be used. Should I attempt to till some of the compost in at the same time or wait a few weeks to add that?

Sorry if these questions sound stupid.....I'm trying!!!