I have a custom object which is related to contact. i am getting the following detail page for the custom object
i am not being able to understand at all where the new Opportunity and new Contact buttons are coming from. They are not lightning quick actions or global actions. can anyone tell me from where can i remove it.
As per my comment, make sure you VF page is marked as Available for Salesforce1 and Lightning. Then the second thing you need to do is override the Salesforce1 and Lightning Experience Actions section in your page layout and add your button there instead of the normal place. You will will find your button available in the Salesforce1 Actions on the top of your screen. The order of your button list will define how it will be displayed in Lightning.
Activity buttons and they are more like tabs in LEX and will stay in the Activities standard component - you should see it in the app builder:
Again - this is the Activities standard component - you cannot change it so you cannot remove the create new textbox and the add button.
Case Comments is a related object list - you can put it wherever you want with the app builder - just drag the Related List component and then select the object:
Best Answer
You should check your custom object layout;
If the red highlighted area as above screenshot, then click there, it will be as below )or it's already as below.
Then you can drag&drop the buttons away from the area.