Controversial! Well, the basic answer to this is that with a couple of very minor exceptions (e.g. the percussion intro to Spiralling, the synth noises and filter sweeps in Everybody's Changing), they are actually playing it all live. In the case of Spiralling, the bass that you're hearing is actually synth bass rather than bass guitar - Tim is playing it with his left hand on the Korg Triton Extreme (the larger synth on the bottom of the stand facing the audience) while playing the piano parts on the Nord Lead 3 with his right hand. Jesse is playing other synth parts (such as the pad sounds and VCS3-style noises) on the Minimoog Voyager. Their synths can be set up to have a split point on some songs, meaning that different keys can be producing entirely different instrument sounds - this is notable on the chorus of Bedshaped - Jesse plays the synth bass with his left hand on the lower keys of the Minimoog, while playing the overdriven pad sounds with his right hand on the upper keys.

They aren't entirely away from the technology though, as it does still provide a click track for a number of the songs, as well as triggering the effects changes for Tom's guitar and changing patches for Tim's and Jesse's synths at the appropriate points in certain songs. This is presumably to allow Tom more freedom to move around with the guitar rather than having to worry about pressing pedals at the right points, and simply because it would be extemely difficult for Tim and Jesse to cope with changing patches for their synths at the same time as playing parts with both hands!

But, in answer to the original question, yes - to all intents and purposes, pretty much everything is being played live.

In the live versions of the song you mentioned during the Hopes And Fears tours, my understanding is that Tim did just use the Rhodes electric piano voice on the Yamaha S90 (note that it's not the ES version). - he used his real Fender Rhodes piano in the studio. As far as I know, the use of the sampled Rhodes sound via Kontakt happened partway through the Under The Iron Sea touring cycle, when the second PowerBook appear on the racks. There are many possible sample libraries that the could have got the Rhodes sound from, so I don't know the answer to the last part of the question at the moment - if I can find out, I'll update this answer.

The live set-up has now been significantly changed (there is no playback from a laptop on-stage), so this is historic information. During 2003 and up until mid-2004, Tim used a MOTU 828 audio interface for the output from the laptop - after this point, he switched to the newer MOTU 828 Mk. II interface. During the Under The Iron Sea tours, there were two of these 828 Mk. II interfaces in the rack - presumably one from the laptop that runs Logic and provides playback and another for the second laptop that ran the Kontakt software sampler. As far as I'm aware, the output from the playback laptop went straight to the sound-desk, so there were no speakers or amps used. The second laptop ran through a Pod XT Pro - again, I'm presuming it then went straight to the sound-desk.