Exactly. And the depots are getting overhead line (strictly it'll be rigid overhead) in the stabling sidings to ensure that the units go out of the depot with their batteries fully charged. Both depots are "islands" of electrification - in the case of Taff's Well there's no wiring in the line connecting the depot to the main line, and in the case of Canton there'll be no wiring between Canton and Cardiff Central or Ninian Park. The FLIRTs have diesel engines but I'm pretty sure TfW are trying to minimise use of diesel in urban areas, and particularly in stations.
No. The Class 398 trains on the Aberdare, Merthyr and Treherbert lines don't have diesel engines, but they do have batteries. The Class 756s have both diesel engines and batteries, but should have sufficient battery power to avoid using diesel through the city centre (and possibly further - some people have suggested they could reach Penarth and even Barry on diesel). If you watch the
video from the PWI the presenters explain how they planned the network to provide overhead power only where it was necessary and easy to install, because the trains using the route all have onboard batteries.
The trains will actually switch between overhead and battery power pretty frequently - if you look at the map
(PDF file at the bottom of this post) only the lines shown in blue on a white background are live overhead. The blue on a yellow background means permanently earthed sections (i.e. there's a wire there, so the trains' pantograph remains up, but it's earthed and delivers no power), and the black lines mean no wires at all. So a train leaving Queen St for Pontypridd would run on battery to Gabalfa, put the pantograph up and pick up some power, use battery to negotiate earthed sections around Llandaff, stay on power to just beyond Radyr, then lower the pantograph and run on battery to just beyond Taff's Well, then pan up as far as Treforest, then pan down again to Pontypridd. The pantograph actuators are certainly going to get plenty of work! No doubt there will be some teething problems. It's also worth noting that stations that stable trains overnight and don't have overhead (Queen St, Central, Pontypridd, Merthyr, for example) will have shore supplies so a train can be "plugged in" overnight and start its shift with a full battery.