SQL Server Profiler - How to filter trace to only display inserts, updates and deletes to one database?

You can narrow down the list by specifying DatabaseName or DatabaseID in Column Filters:

Note however that you need to check the box "show all columns" first.

As for inserts, updates and deletes you can filter by text column at the same place:

You can filter by database name or database id.

Use SP_trace_setfilter to filter for one database.

sp_trace_setfilter (Transact-SQL)

If you are doing from GUI (which I do not recommend, you are better off with a server side trace), click filter columns and then chose DatabaseName. See the picture below for doing it via GUI: