How much do internships and journal publications matter to AOs, as long as they are related to your intended major? Say in comparison to maybe, academic achievements (clearing xyz stage of some Olympiad/getting a 98% in 12th boards compared to 96%)

College AOs follow a holistic approach to admissions. They are looking for strong academic students, who are committed and passionate about something. It does not have to be academic but it should demonstrate achievement. The single measure for a lot of this is a level of achievement which shows focus and success.

Internships in the area of your passion/interest establish your commitment to your area of interest. Clearly getting research published in a major journal is a big accomplishment, as are winning selective awards like Intel Awards.

In the end, your application is a total package and how it stacks up individually and vis a vis other applications determine the outcome.

Adding onto the previous comment, I will say that if going from a 96% to a 98% is going to take up a significant amount of time that you believe you could've spent writing a research paper then just do the research paper.

However, if you're debating between working hard to go from an 85% to a 95% or working on a research paper then try to get to 95% because colleges want to make sure that first and foremost you are able to excel in academics and that you are then also able to do other stuff outside class, but academics comes first because they want to make sure you will do well in college classes.

Olympiads are always great because they show AOs that you are taking that extra step and that you are indeed more academically accomplished than someone who just focuses on school work but again, olympiads should only be in addition to great school work and not at the cost of it.

Getting independent research published is an extremely impressive accomplishment especially if it is linked to (or is itself) even remotely groundbreaking - this is often impossible to achieve but nonetheless independent published research is a seriously impressive accomplishment - even co-authoring papers is a significant accomplishment so if possible it is something you should certainly try to do as it is a significant indicator of intellectual vitality and true passion for a subject.