Absolutely without a doubt take it to a dealer assuming you still have the factory warranty. Make sure you have the proper amount of oil in there and put in what the oil cap says to as far as viscosity and that the oil meets the minimum Ford Specification. Too much oil can be harmful too.
I don't know if 87 octane is OK per the manual. I use 91 minimum and try to use 93 whenever possible.
You probably already know but the rear end shake is called wheel hop and is common in wet conditions especially with stock suspension.
Good Luck I hope it is nothing. However an engine knock has to be something.
2013 Mustang GT Premium, Deep Impact Blue, GT Track Package, Recaros, Ext. Acc. package 1, All Weather Mats, GT500 Axle backs, Airaid CAI, BMR Rear LCAs, BMR Relocation Brackets, JLT Oil Separator(s), Barton 2 Post Shifter Bracket, Steeda Motor Mounts, Barton Short Throw Shifter, JHR Clutch line, BMR adj. UCA and Mount, Kooks 1 3/4 headers with catted H, DSS Al Shaft with BMR Loop