Landing Pages

Yes, on the General settings tab you can choose stores where your landing page should be displayed:
...

Yes, you can change landing pages. There are two fields – Top Description and Bottom Description – where you can insert text that will be displayed on your landing page....

There are 2 possible algorithms:
1) use magento default attributes "new from" and "new to"
or
2) set the new period in the module setting and use product creation date....

Yes, there is such a possibility. On the Conditions tab you can choose which categories and/or subcategories should be included:
...

Yes, you can use the Advanced Filter on the Conditions Tab to achieve that:

As a matter of fact, you can use any product attribute to add items to your landing pages....

Yes, you can easily do it. Say, you need to display products with a price of $5 - $20. In this case you should add multiple price conditions like "price greater than $5" and "price less than $20" as well as multiple attributes conditions:
...

Our Landing Pages module works well with the Flat Categories and Products option. But in case there are still any issues our support managers will fix them for free....

Yes, sure. To do that please navigate to Admin Panel > System > Configuration > Landing Pages and switch configuration scope to website. Once you do that, an import option will be enabled:

 ...

Our Landing Pages module is fully compatible with default Magento navigation and our Improved Layered Navigation....

If you need to change sort order, please paste this code into the “Layout Update XML” field on the Design tab:

<reference name="amlanding_product_list_toolbar">
<action method="setDefaultDirection"><string>desc</s...

Please paste the following code to the Layout Update XML field on Design tab:
<reference name="head"><action method="setRobots"><meta>NOINDEX,NOFOLLOW</meta></action></reference>...

If you get a “There are no products matching the selection” message on the landing page, please check the settings of the root category. For the correct work of the Landing Pages extension “Is Anchor” option of Root category s...

To fix it please navigate to Admin Panel > Catalog > Attribute > Manage Attributes > select the needed attribute there and set the “Used in Product Listing” option  to “Yes”. Please note that re-indexation is ...

If you mean using slash signs to add categories and/or subcategories to the URLs, like /promo/on-sale/products.html, then it's not possible with our extension.
 ...