#1
I know how to palm mute when playing with a pick. And I know how to play muted passages fingerstyle using my thumb and index finger. But how should I handle a fingerstyle song that mutes some notes as accents? For example, Machine Head by Bush. Should I just play the whole song with my index and thumb and just drop my palm to mute?
#2
Listening to the song you mentioned now. Firstly, that's almost definitely played with a pick anyway. Secondly, it sounds like dead notes rather than palm muting, which you can get by simply muting the strings with your left hand (resting your fingers against the strings but not fretting anything). Moving the position of your left hand will change the tone, it's best to try and avoid any strong harmonics while doing this (i.e. muting over the 5th, 7th of 12th fret).

If you do want to palm mute with a fingerstyle technique (which is unlikely what's happening in this particular song), it's possible to do so as you mentioned by resting your palm against the strings and plucking with your thumb and fingers. Another way is to use a more conventional bass fingerstyle technique, but play fairly close to the bridge and rest your little finger of your plucking hand against the string you're playing to achieve the same sound as a palm mute (even though you're muting with the tip of your finger). This can be quite uncomfortable at first and will take a bit of getting used to.
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