Converting differential-current output to single-ended voltage; adjustable gain
Can I do it with following differential op-amp configuration at the output of the DAC?

If that configuration looks like figure 4 of the DAC datasheet, then yes.
(hint: it isn't. You want to have a defined, power-matched termination to ground; remember, you're not using a 125 MSPS DAC to produce DC; so you have to work like someone who has high-speed analog, not just a nearly constant signal.)
But: the AD712 has maybe 3 MHz of gain-bandwidth product. You're feeding that with a 125 MSPS DAC. Either you've chosen a DAC that's totally overkill for your application, or the opamp is far far far too slow.
I'd simply go and read the datasheet of the opamp recommended in fig. 4. It's not a random recommendation.

I need to make the output voltage gain adjustable; how can that be done?

By using a variable gain amplifier (VGA) or programmable gain amplifier (PGA). These are meant for exactly this use case, adjustable amplification.
But honestly, if you have a DAC and want to adjust the amplitude of the output, do it in the DAC.
Your DAC even has a dedicated gain setting functionality. So, really, read the DAC datasheet from front to back until you figure out how to use that.
If that isn't fine enough (or large enough) in dynamic range, you could also consider scaling the digital values.
If it can be avoided, don't try to multiply analog things; that is very hard to do accurately. That's why DACs are a bit more expensive than they'd be if they were just a bunch of resistors and transistor switches thrown in an IC. Let the DAC do your analog signal scaling – it's what you've bought it for.