Yes, you can do a RESTART from a SOL111 run into a Random run... this allows for changing the data recovery requests from one run to another. in fact, the attached sample will run normal modes, restart into frequency response, and then restart into a Random analysis...
Regarding the solution restart for the random analysis. Can I use stress(plot) in the modal solution to include the stresses in the database for restart?
I'm not sure of the question... if you use STRESS(PLOT) in the SOL111 run, then the frequency response stresses will be stored on the results file (op2, xdb, etc), but NOT printed to the f06. This doesn't have any bearing on the Random portion of the response.
If you are looking for Random results, then you will use the RALL (or just PSDF) tag inside the data request... such as STRESS(PLOT,RALL)=ALL... maybe it would be best to take this offline to a technical support case if there are still issues to discuss?
No, it doesn't... but it is usually a good idea to get some data recovery at each stage of the process to ensure the results make sense before moving on... such as looking at modes before running frequency response and looking at the FRF from frequency response before looking at random results. You can change data requests from run to run, even though it is a restart.