Replacement surface mount handle set

Your surface mount handle set looks close to our 82-005 with a 82-306 keyed exterior pull. Take a look and go over our measurements. Confirm these two parts are what you have been looking for. 

The 82-005 handle set comes with screws. As for the keeper, I can't say what you need. The 82-005 is compatible with a couple different kinds, like the 89-012. If the old keepers are in good condition you can just reuse them, but if they aren't then we won't be able to offer a suggestion without seeing a photo of one.

Lonita has a surface-mount handle set on her door; these are very versatile hardware sets since they use their own internal locking mechanism. They don't rely on anything in the original door whatsoever, allowing them to install on just about anything.

The downside to this is since they mount on just about any kind of door, it's not really possible to predict everything you might run into when installing it. I am confident that the  82-005 will replace the handle set I see in your photos, but aside from that we can't really go into more detail. If there are any tertiary parts that you may need, or that may cause a problem, then we'd have to see photos of them all one-by-one.