Will Trigger and Workflow get fired while using Bulk API

Triggers and Workflows will fire irrespective of the API you are using. Form the link that you have shared please go to the section:

Triggers
Although the flexibility of triggers is a great asset, triggers can cause many kinds of problems in your loads and integrations. Locks are one of those problems. When you’re loading records that fire one or more triggers, and any of those triggers performs either a select for update or a DML operation on records other than the record that you’re inserting, updating, or deleting, Salesforce locks those other records. These locks can in turn cause lock exceptions. Consider disabling trigger logic for your loads and integrations, or having a special, optimized code path for such activities.
Workflow Rules
When workflow rules trigger field updates, Salesforce locks the records that they update. And when multiple threads try to update the same records simultaneously, they can cause lock exceptions. Consider defining your workflow rules so that they don’t execute during loads and integrations.

Basically it implies that logic should be efficient enough to handle large data sets(Batch size).

Did you try adding a new trigger or workflow?I have checked it by creating new work flow rule that updates Account number whenever a custom field called kunnr(SAP Customer Number) has been inserted through bulk API.It works perfectly.

Workflow rule:

And imported around 2500 records via bulk api using curl.

Result:

here SAP Customer Number is kunnr

It worked for all the records as such..