Q1) Can I add the gift directly to the cart when the customer add the eligible product?
Yes, enable auto add function when creating the offer.
Q2) Can I set the options inside the Gift? Like Pillowcase color, small size, etc?
Yes, gift products are like your normal products. You can set up options.
Q3) Does it support custom theme like journal, fastor, pav theme, etc?
We have configured our extensions mostly on all custom themes. There are no extra charges for the configuration. Sometimes the extension works without configuration and sometimes configuration is needed. Depends from theme to theme.

For configuration, you can create a support ticket within one year from your purchase date.
Q4) Is it possible that Free gift can be visible on shop can sell separately also??
No. Free gift can only be seen on the shopping cart page. And they are active only when conditions satisfy.

If you want to gift products to be purchased separately then check the extension Product combination offers on our store:

http://www.opencart.com/index.php?route=extension/extension/info&extension_id=25098
Q5) Can free item be added automatically to the cart?
Yes if you enable auto to add for the gift, then it would be added automatically. If your free gift has options then you should not enable it.