Will the Pc1x1 connection limit the speed of this Device

Astrotek PCI Express SATA III 2 Port Low Profile Card [AT-CPES6A] - $26.00 : PC Case Gear
Also will this card be limited to PCIex1 speed if plugged into a PCIex1 motherboard slot
ASRock SATA III PCIe Card [AR-SATA3-CARD] - $29.00 : PC Case Gear

It really depends on the drive(s) you are connecting to it. The PCIE-1x slot if (1.0) will have a single direction max rate of aprox 266MB/s and if (2.0) 532MB/s.

This will be fine for 7200rpm Hard Disks but if you were planning on one or more 500MB/s SSD's then the bus-width would limit you.

The devices only claim to be compatible with a PCI-E x1 (2.0) interface.

It's confusing because the Manufacturer claims that if you use a PCIex1slot you can obtain PCIex2 speeds (sata3),is it possible the card boosts the speed

Please link to where it claims this?

It has 2x Sata III ports on it. But cannot transfer data across the board faster than the max speed of a single PCIE lane.

Depending on the controller on the card, two drives using it may well be able to transfer at full speed between each other.

EDIT: It wouldn't be the first device to be sold with a bottle neck.