How deep do I dig to install a walkway when there is already a trench that has some vegetation?
You don't need to go deeper than what you need for your design.  Remove any substantial exposed roots on the starting surface (say pencil diameter or larger), mainly to level it and prevent voids if they eventually rot away.  Landscape fabric can't hurt, but I'm not sure how much difference it will make.  

It's generally a good idea to start with a compacted layer of paver base.  The material is a designed mix of sizes, and the pieces aren't rounded.  It interlocks to make it more stable so it doesn't shift and settle when it gets wet.  The finer material gets washed into crevices between larger components, making it even stronger and tighter.  Once it's compacted and gets a little water, it can approach concrete in its characteristics.  It also helps to block roots.  Then use a layer of sand for the slabs.

Be aware that your decorative stone and underlying sand will quickly form a "garden".  Dirt will collect in the stones, and seeds and pollen that fall there will grow (even if you have a layer of plastic directly underneath).  You will need to regularly treat it with weed killer.  I've done a couple of approaches to create a maintenance-free barrier.

One is to use exposed aggregate concrete as the filler where you're showing the decorative stone.  You can mix in colorful pebbles and then wash the surface to expose them.  If you want to stick with a look closer to what you show, you can fill partially with mortar or grout and then embed a thin layer of stones in the surface.  

Another approach I've used is to put in the decorative stone chips, add some fine sand to fill the lower voids (wash it in until you can just start to see it), let it completely dry, and then add plastic resin.  That amount of resin can get expensive, but you can make your own cheaply.  Explaining how to do it is a bit long and tangential to this question, but if you're interested, we can dedicate a separate question to it.