Can I use the existing drives (Data,log) drives, create a new database and map the mdf and ldf to the existing mountpoints?
Yes you can use the existing drives for new database. You would need a new set of shared drives if you are installing new SQL Server clustered instance.
Do I need to create another MSDTC drive for this application?
Not necessarily, a single MSDTC can work for multiple instances. But if you want to know what all queries need MSDTC I would direct you to Cindy Gross Blog Do I Need MSDTC for SQL Server Cluster. Quoting
If your application calls DTC directly or if you use a feature in SQL Server that calls DTC, then you need DTC to be available. Some examples of how SQL Server uses DTC: linked servers, OPENROWSET, OPENQUERY, OPENDATASOURCE, remote procedure calls, BEGIN DISTRIBUTED TRANSACTION, updatable subscriptions for transactional replication (immediate and queued updating are now deprecated). You can choose to enlist in a DTC transaction from CLR, SSIS, and DTS. SQL Server does not support using DTC with some features such as database mirroring.