Precision 5810 PCI lanes, and bifurcation configurations. Need help!

For the adapter, if you want to keep the future upgrade option open, get the Asus Hyper v.2.  You can use with just one NVMe drive for now and upgrading more, up to total of 4 NVMe drives in the future.  If you are certain that you will be happy with just 2 drives, get the ROG Hyper.  Either one are passive adapters and will provide similar performance as Dell Ultra-Speed adapter.  They do not affect the speed of the SSD.

For the NVMe SSD, Crucial P3 Gen4 will be limited by the PCIe Gen3 slot and expected max performance of 3500 MB/s.  

For PCIe lanes, your system has 40 PCIe Gen3 lanes direct to CPU, and 8 PCIe Gen2 lanes from C610 chipset.  Your graphics card is occupied on one PCIe x16 slot (CPU), that leaves the option for NVMe adapter to use the second PCIe x16 slot (CPU).  Your Wifi card uses just one lane from PCIe x1 (PCH).  You still have one empty slot of PCIe x8 (CPU), and one PCIe x4 slot (PCH).  Note that chipset (PCH) lanes are PCIe Gen2.

Glad to help.  Using the picture numbering, then the slot numbers ( ) are imprinted on motherboard:

#1.  It's a PCI slot for legacy (old) add-on card, don't use it for your storage upgrade.  (slot 6)

#2.  Is a PCIe x16 Gen2, wired as #4 lanes, reserved for Dell thunderbolt 2 add-in card.  (slot 5)

#3.  Is a PCIe x16 Gen3, full bandwidth, use this slot for your Asus ROG Hyper adapter (slot 4)

#4.  Is a PCIe x1 Gen2, this was suggested for your Fenvi Wifi adapter, but use #2 (slot 5) instead. 

#5.  Is a PCIe x16 Gen3, full bandwidth, mainly use for graphics card, MSI Vega 8 GB (slot 2)

#6.  Is a PCIe x16 Gen3, wired as x8 lanes, may use for future upgrade (slot 1)

Gen3 are slots connected to CPU, Gen2 are slot connected to PCH (chipset).