You have not pasted an example of your incoming email, so I will make some assumptions. Can you confirm if the emails are generating Incidents at all?
In your mappings, I see you have mapped fields like Category, Impact, Urgency to the BODY.
If the BODY has the delimiter on the Email Keywords, then only it will work.
As far as 3rd party log number is concerned which is mapped to Subject field, with the Call Logged - Incident as email keyword, you wont be seeing any mapping.
Hence I suggest to do the following:
1. Clear the "email keyword" field for 3rd party log number field.
2. It will dump the entire subject line in incident creation.
3. Add a work flow with field update which will extract the actual number from the subject line.
Workflow rule criteria If 3rdPartyfieldNumber != ''
FIeld update formula:
Then Update the 3rdPartyNumber field with a formula.
Formula: SUBSTITUTE( 3rdPartyfieldNumber__c, 'Call Logged - Incident: ', '')
Basically, the formula will replace the text 'Call Logged - Incident: 'with nothing leaving just the number in that field.