What should someone who recites the Quran with Tajweed do if their voice bothers others and they describe it as "detestable"? Should they recite the Quran and adhkar (remembrances of Allah) in their heart without sound, or should they ignore what they say?

The sufficient amount of recitation that earns reward is moving the tongue and lips and articulating the letters from their proper articulation points, while making oneself hear it. It is not a condition to raise the voice in a way that disturbs others; rather, a sound that you can hear yourself is sufficient, along with observing the obligatory (not merely recommended) rules of Tajweed.