> 1. Have we detected such lobes in the milky way? why not?
Sag A* (our black hole in the center of the Milky Way) isn't considered "active" right now. We don't notice it gobbling up stars and gasses, which would be necessary for the jets to be possible. I remember back in 2020 or 2021 there was an article that we're noticing a jet from Sag A*, which we're still trying to understand why because we don't expect Sag A* to be active. It's also super difficult to monitor Sag A* since there is so much dense dust, gas, etc in the way between us and the SMBH.
> 2. If those particles are going near the speed of light yet have no reason to slow down unless captured, unlikely outside of their original galaxy, they are still going for billions of years?
Generally speaking, yeah, they are! If we're looking at photons though, they will eventually get red-shifted so much that they'll become infrared (invisible to us), until their energy is so low that it'll be near impossible to see without telescopes more powerful than anything we have right now.
> 3. If some of those jets from other galaxies are pointed at Earth and contain physical particles with mass near the speed of light, why don't they do measurable damage?
Space is a vacuum, but there are still things that can slow these particles down (loss of energy like photons, gravity wells from other massive objects, running into a spec of space dust, etc. Also, space is very empty, and statistically, it's incredibly improbable that one of these jets could be aimed directly at us, while also being close enough to us, to cause damage. We do notice them though! They're powerful enough to get picked up by scientific instruments, but are not concentrated enough or powerful enough to cause damage to us or Earth.