The Toshiba NVMe SSD is Gen3 x2 , while the 2013 iMac is Gen2 x2, so there shouldn't be anything limiting the performance. In 2013 27" iMacs people achieve about 750-780 MB/s read/write.
This is an interesting adapter you have bought. It is cheaper than the SinTech and supports NVMe, at least according to the title. However, it looks just like any other adapter that is listed for AHCI SSDs-only.
To fix the waking problems, go to System Preferences -> Energy Saver and untick "Wake for network access" AND/OR "Enable power nap". For me, I think that network wake alone fixed it, but I have the power nap off as well.
It is very interesting that your machine sleeps and wakes normally. Do you have OS X installed on the new Toshiba NVMe or on the original 3.5" HDD (i.e. which drive is the boot volume in?). Also what OS X are you running?
Can you provide the following information from Terminal:
Power -> sudo pmset -g
EFI -> /usr/libexec/firmwarecheckers/eficheck/eficheck --integrity-check
boot rom -> system_profiler SPHardwareDataType | grep -i "Version" | awk -F ':' '{print $1 $2}'
The wake problem what everyone is experiencing is caused by waking from sleep when hibernatemode is different than 25 and when standby is different than 0. If OS X is installed on the NVMe these are the below scenarios:
1) hibernatemode = 0 or 3, standby = 0 -> upon trying to wake, the Mac will reboot with a message "Your computer restarted because of a problem"
2) hibernatemode = 0 or 3 or 25, standby = 1 -> upon trying to wake, the Mac will show a folder with a question mark in it, and you will have to long press the power button to shut down your computer
3) hibernatemode = 25, standby = 0 -> the only option for a successful wake
autopoweroff, sleep, displaysleep, disksleep settings do NOT matter.
The above issues occur because the bootrom contains an archived inactive version of the NVMe dXE driver. IMacs from 2013 onwards SUPPORT NVME, but we would only be able to make our NVMe work properly if we are able to extract that NVMe dXE driver, or embed it from a 2017 iMac's installer and replace it on a software level (flash it to the motherboard's SPI rom chip), just like OS X would otherwise do if it would recognize a newer 2017 iMac.