Does it connect to wifi?

Yes, it has LAN 10/100/1000Mbps and Bluetooth 4.0 for connectivity.

I have the same question is the disk 128GB as listed in the specification or is it 512GB as listed in SSD and product headline?

The disk storage is 512 GB SSD as listed in the product headline and SSD specification.

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Dell Latitude 7000 Series (7440) 14" HD LED Intel Core i5-4300U 16 GB DDR3 SODIMM RAM 512 GB Solid State (SSD) HDMI Webcam Windows 10 Professional