I agree with the highly rated answer but thought I'd add the following:
* Do you use a staging area to perform the transformation and then
load into the warehouse?
It depends on the type of transformation whether it will require staging. Staging offers benefits of breaking the ETL into more manageable chunks, but also provides a working area that allows manipulations to take place on the data without affecting the warehouse. It can help to have (at least) some dimension lookups in a staging area which store the keys from the OLTP system and the key of the latest dim record, to use as a lookup when loading your fact records.
The transformation happens in the ETL process itself, but it may or may not require some staging to help it along the way.
* How do you link data between the warehouse and the OLTP database?
It is useful to load the business keys (or actual primary keys if available) into the data warehouse as a reference back to the OLTP system. Also, auditing in the DW process should record the lineage of each bit of data by recording the load process that has loaded it.
* Where/How do you manage the transformation process - in the
database as sprocs, dts/ssis packages,
or SQL from application code?
This would typically be in SSIS packages, but often it is more performant to transform in the source query. Unfortunately this makes the source query quite complicated to understand and therefore maintain, so if performance is not an issue then transforming in the SSIS code is best. When you do this, this is another reason for having a staging area as then you can make more joins in the source query between different tables.