Best thing to do is to write your own CSV import php file. By using functions like wp_insert_post() you can insert the data from the csv file as a new post in the wordpress database.
If you’re not a coder, which I’m sensing, you can try this plugin: https://wordpress.org/plugins/wp-ultimate-csv-importer/.
The free function offers everything you’re looking for. The pro version offers more in terms of custom fields support. But assuming your custom fields are default wordpress custom fields, the free version will do.