My suggestions, with comments in parentheses:
Där succé rimmar på plocka ner honom - where success rhymes with "take him down" (i.e. where people are jealous and vindictive)
Och helgonen hade åkt fast för fylleflygningar - and the saints had been busted for flying under the influence
Och ingen bad längre, nej de bara predikade - and no one prayed anymore, no, they only preached
Fick motorstopp på fel sida av himlen - engine stopped on the wrong side of heaven (no police presence implied)
Han sa: "Vi drar, det finns ett grymt universum här intill - He said: "let's go, there is a wicked universe close by" ("grym" mean "cruel" but is used in a positive sense, like "wicked" in English)
Jag var inlåst i en pyramid, i en evighet med ett hundlik - I was locked away in a pyramid, for an eternity with a dog corpse ("hundlik" as an adjective means "doglike" but here it is a noun, as witnessed by the indefinite article)
Jag ska bara skriva världens bästa avskedsbrev - Just let me write the world's best farewell letter first ("jag skall bara X" usually means "I'll just do X first")
Och jag kände mig som månen, ärrad och avlägsen - and I felt like the moon, scarred and distant
Och jag visste, att dom visste, att jag visste, att vi var på fel väg - and I knew, that they knew, that I knew, that we were heading the wrong way
Men vi bara fortsatte, vi skulle hela vägen in i kaklet - but we just kept going, all the way to the end ("hela vägen in i kaklet" literally means "all the way into the wall tiles" but is an idiom indicating that something is done with conviction; presumably from competitive swimming, a contestant giving 100% until touching the wall)
Oden ville bli påve i Colombia - Oden wanted to become pope in Colombia
Min vänstra liftade tillbaks till jorden - my left (hand) hitch-hiked back to Earth ("att lifta" - "to hitch a hike" although maybe in your variant of English "lift" covers this, in which case my change is unneccessary)
Jag lämnade dem där, nedrökta på buddhism - I left them there, reeking of Buddhism ("nedrökt" mean "permeated by smoke, showing signs of extensive smoking" so the image is one of potheads going on about Buddhism)
Vad som var tomt i mig, skulle bli musik - what was empty in me, would become music
Om det inte funnits en bit GBG, ja du vet - (BGB is short for Göteborg/Gothenburg, Hellström's much-mythologized home town)
Förslag och korrigeringar är alltid välkomna.