How do I create a new mongo collection with custom indices from an existing mongo db?
Your title and question have slightly different goals (copy vs. modify).

To copy a collection between databases you can use the cloneCollection method:

https://docs.mongodb.org/manual/reference/method/db.cloneCollection/

Once you have a collection in the new database that you want to modify you are able to create all the indexes you want to the existing data.

Some of the considerations to be aware of based on the larger data size of the collection is whether you want to block other activities for the creation of the index.  By sending the creation to the "background", you can prevent this:

https://docs.mongodb.org/manual/reference/method/db.collection.createIndex/#options

The other issue, you might want to be aware of, is that index builds in the background generally take longer to build.  One option is, presumably if you are running a replica set, is to build the index on one node at a time, after you remove each node from the replica set.  You can build the indexes once the node is out, and then put it back into the replica set, and move on to the next node.

As far as removing unnecessary fields, you can do that as you go, and just process through the data in batches.  Even if you do it in one query, if it dies, MongoDB will still maintain all the data you already updated, so it can pick up where it left off removing fields that exist in the remaining documents.