None of the sentences work for me. I would use "despite" or "in spite of".
1) With a bad work record, we still hired him.
2) We hired him with his bad work record.
3) We hired him, with his bad work record.
Are these sentences correct?
'With his bad work record' is supposed to mean 'in spite of his bad work record' in them.
As far as I know, 'with' can be used instead of 'in spite of', if it is followed by all.
4) With all his efforts, he did not manage to get the job done.
You could use even with his bad work/employment record.