Should I launch my product everywhere on Product Hunt and Hackernews and IH on the same day? or day by day?

I think yes. I launched like this (San Francisco time):
- PH 00:01
- HN 05:00 - 10:00 AM. You can try to post on HN just in couple of hours if you will don't get traction but with different title.

great, thanks

I usually post on HN at around 7-8 am San Francisco time, just when the newest page starts picking up pace.

For I launched everywhere on the same day.

First on PH (due to the midnight timing.)
Then on HackerNews (who just flat out hated on me)
Then on IndieHackers (who loved it)

I then went on to do some research regarding other platforms to launch on after implementing some early feedback from the above platforms and finally launched on DesignerNews. (Which got me featured in all sorts of places)

So I think PH, HN and IH are good to gather initial feedback and see if anything is still rough around the edges and then you can deliver a semi-polished product to your niche. :)

That's a great question. I tend to launch on most platforms the same day, because I feel like they value their posts being of new product rather than a week old. However, I haven't really tested this nor can back it up with data.

The benefit of a staggered launch with a few days in between is that you have time to incorporate feedback from the first launch into the next, and so forth.