Are ESXi boxes manageable from VirtualCenter right out of the box? Or do they need a license added? Or do they also need some software added?

You don't need to add software,  but  an ESX Single Server Partitioning license (http://www.vmware.com/products/vi/buy.html) doesn't included a VirtualCenter license.  Thus you would need to purchase / upgrade to  at least ESX Foundation or higher licensing.

Are ESXi boxes HA-capable right out of the box? Or do they need a license added? Or do they need some software added?

You would have to get ESX Standard licensing and  you would need to purchase a VirtualCenter license.  When you add an ESXi host to VirtualCenter or enable it for HA, configuration change are made on ESXi and software is added (it's automatically added).

Same for VMotion and DRS. Do I need an upgrade license? And are any software changes required?

You would need to upgrade to ESX Enterprise license and have a VirtualCenter license.

>Would I ever need to replace ESXi with ESX on a server?

If you start with ESXi and are happy with it, no you would not.  ESX regular will be phased out eventually and ESXi is the product going forward.   ESXi can be a difficult change from ESX if for example you depend on service console scripting (you have to look at alternatives for ESXi) or rely on softare that isn't supported on ESXi yet.

Are the capability-related answers the same for ESXi Embedded and ESXi Installable?

Yes.