How to produce high-resolution DEMs from new Sentinel-1 data?

I'm guessing you never worked with SAR data before, so I'll break your question down into parts I can answer:

1) Create high resolution DEMs in GIS

The process of creating a DEM just from SAR data is quite complex and requires a lot of processing power and memory. I don't know of a GIS software that implements DEM creation due to these constraints.

2) Investigate sites/landscapes and model their change

If you want to model/observe elevation change Differential Interferometry is the key word you should be looking for. This approach enables you to observe even small changes in elevation without the complex process of creating two DEMs.

3) How to produce high-resolution DEMs

This is a complex process requiring a lot of steps. I highly suggest you read up on SAR Interferometry in general, since Sentinel-1 data has basically the same (dis)-advantages as any other SAR data.

ESA provides a free book containing all relevant information regarding SAR interferometry, be it from a mathemathical as well as a processing viewpoint:

ESA: InSAR principles

Steps required to create a DEM

• co-register scenes
• calculate baselines
• create interferogram and extract phase information
• correct for sensor movement, flat earth removal, athmosphere, etc.
• convert differential phase to absolute phase differences
• phase unwrapping
• phase to height conversion
• correct for absolute elevation, phase jumps, etc.

This is obviously just an outline. The detailed processes for each step are explained in the book I refered to and, in theory, can all be executed with the S1-Toolbox (soon to be SNAP) or the Next ESA SAR Toolbox (NEST).

Here I found an article "Preliminary results of using Sentinel-1 SAR data for DSM generation", 2015. As I understood, the vertical accuracy will not be better than 20m (an average height of a single tree). That is comparable to the SRTMGL1, GDEM2 accuracy.

I didn't find it useful to improve the SRTM DEM, by filling the forest masked areas in the SRTM1GL (to get rid of the tree cover vertical offset). I am not a SAR, DEM-specialist, though.