Linear time algorithm for finding $k$ shortest paths in unweighted graphs
No. The best times known can be found in the survey "Implementations and empirical comparison of k shortest loopless path algorithms".

As a note, it is impossible to get linear times in the weighted case. There are lower bounds for the shortest path (i.e., k=1), which are not linear.

In the directed acyclic case you might have linear times for the k-SP, but I think is an open problem.

Cheers