Delayed posts

1. Not by default. You can ask them for a date, store that value, and change the post status to scheduled using that date. Or, there might be a plugin at the WordPress.org repository that will allow you to schedule posts like that, using a date stored in a custom field. You can collect the data with Gravity Forms (the post date) but you will have to do the work to ensure it shows only after that date.

2. Not built in. But you could do it using the hooks built in to Gravity Forms. You'd could use custom validation (there is a filter gform_validation) and in that, check for how many posts were already made, in whatever time frame, then return a validation error. Or, you could run something before displaying the form, to make it unavailable once your limit has been reached. It can be done, but it's not built in.

3. The date picker is the default jQuery date picker, and it does not have that functionality. You can disable certain dates, but it does not have a concept of a filter for available dates, I don't think.

I don't think Gravity Forms will do all the things you need, right out of the box. You will need to do some work to accomplish them. Some of the things you describe are fairly complex customization and would require a fair bit of code. I hesitate to say that Gravity Forms will work for you without knowing more about your PHP knowledge and willingness to dig in.