Is DIY water softener feasible with my house layout?
Yes, a DIY is feasible, provided you don't want whole-house soft water, just soft "hot water".  The valve is for your cold water intake (a safety feature that should be on all water heaters), and you'll cut that line and rework it into your water conditioner, taking the outlet of your conditioner back into your water heater cold inlet.  You'll keep a cold cut off valve (but probably not the same one), placing it between the wall and your water conditioner.

If you want your cold water to be soft too, you need an exterior softener between the house and the whole house cut off valve.  I'd recommend avoiding DIY for an exterior job, as fixes are more complicated if things go wrong. 

Even though it is feasible, there are many reasons you might not want to do this job:


It's water, and good PVC joints are made with dry pipe.  This means a lot of drying is involved, and rework of mistakes take a long cycle of shut off water, remove, dry, add, turn on water, test (and hopefully not repeat). 
Some jurisdictions permit water heater work as DIY, some require a permit, some require a licensed plumber.  It's not fun getting fined if you are discovered to be on the wrong side of your city's, county's, state's code.  Permit doesn't mean no-DIY, but it does mean committing to a DIY that passes code.
Should things go wrong, you don't have the convenience of someone else's insurance / wallet / time to effect the repairs.  One example of something going wrong might be breaking the in-wall pipe attempting to rework the cold water line.  Depending on where the break is, you might be DIY'ing drywall too.  
This kind of work seems easy and fast, but often it takes more time than expected.


If you've rebuilt a toilet, installed a sink, and fixed an AC drain line, you're primed to replace a water heater, and are primed to do this job.  If you've replaced a water heater, this is not more difficult, but there are more cuts and bends involved.  If this is your first plumbing DIY it's still within the realm of feasibility for some (demeanor, etc.) while it could be a very challenging DIY for most.  If you're on the fence try draining, flushing, and filling your water heater to get a taste of about 1/4 the work.

Good installations have wall mounted pipe, good hoses for where they are required, and a good (tested) drainage system to catch the leaks and spills for canister changes.