Show Field from Separate Table based on Corresponding Field

Yes, the Job# is unique. Only 1 entry per Job # in a table, but the same Job # appears once in each table.

So a suggestion is to change the key field of table 1 to be the Job#.  Then make a relationship where 1 Table 1 has many Table 2 (even if there will only ever be 1 child).

Then you can look up any information from Table 1 down to Table 2, including its completion status. 

Is there a formula I could use or a way for Quickbase to recognize automatically a Job# match?
I created the relationship, but I want to avoid a user-look up to pull in the Table #1 information. The Job# in Table 2 will be imported.

Sorry, I left out an important step.  You will need to change the Key field of your table 1 to the Job # field.  I would suggest doing that and then deleting the existing relationship and building the relationship again.

Then the records will link up automatically if the reference field on the right side of the relationship is the job #.

This worked between two tables, but will this work if there are multiple relationships between the tables? I am going to be making a lot of relationships (see below). Each column is a stage in our production process and each Machine Box is a separate table in our app. Different machines feed the next machine.
Ideally, Machine 5 will have fields saying "Machine 1 - Job Complete", "Machine 2 - Job Complete", "Machine 3 - Job Complete" and "Machine 4 - Job Complete".

Do you think this kind of set up is possible?

Yes I think its possible.

Machine 5 has many Machine 4As.

Machine 4A has many Machine 3s.

Machine 4B has many machine 2As
Machine 4B has many machine 2Bs
Machine 4B has many machine 2Cs

Machine 3 has many machine 2As
Machine 3 has many machine 2Bs
Machine 3 has many machine 2Cs

etc