Ways to Upgrade Production MySQL db server from 5.5.42 to 5.7.13 with zero downtime
1) No. You have to restart MySQL and that means downtime. 

2) To minimize the downtime you need an identical server which you can switch to and back. The procedure on high level:


you can build a passive master using percona-xtrabackup for example without downtime
Set up master-master replication
Upgrade the passive master to 5.6 and wait for it to catch up in replication
Switch active master to be the upgraded one (running 5.6)
Stop and rebuild the old master with the same process you did in step #1 but now from the new version (5.6)
Repeat the procedure to upgrade for 5.7
If you don't need it trash the passive master