Picture size

If you're defining the pixel dimensions you can ignore the DPI setting - it doesn't do anything.

What you mean by "websites", I generally go with 600, 800 or 1200 pixels on the longest edge depending which website I'm going to use it on (smaller for Facebook, larger for Flickr).
 
Just to get the highest quality possible.
Quality has nothing to do with the maximum number of pixels, it's about resizing to the right number of pixels for the job. That may mean multiple exports for printing at different sizes and sharing in different places.
 
This is a minefield. I used to be a computer engineer (in days long gone by) and this pixel density/quality thing still confuses me.
 
To make a high quality print labs need 250-300 pixels per inch. 9933 pixels would take you towards a 40 inch print.
 
This is a minefield. I used to be a computer engineer (in days long gone by) and this pixel density/quality thing still confuses me.
It's easy..

(Print dimension in inches) x 300 = the number of pixels you need

Assuming 300 dpi/ppi which is fairly standard
 
It's easy..

(Print dimension in inches) x 300 = the number of pixels you need

Assuming 300 dpi/ppi which is fairly standard

The bigger the print the lower the density you really need.
an A4 print in the hand can be examined minutely
a wall sized poster must be seen from a distance.
 
The bigger the print the lower the density you really need.
an A4 print in the hand can be examined minutely
a wall sized poster must be seen from a distance.
But up to A3 it's a fair generalisation..

Not many people print at all, never mind to wall-sized poster size.
 
Most labs will render the image themselves anyway, including resizing the image.
Some printers size based on the DPI and the setting assigned, therefore we need to render the image and resize it to make sure it gets printed the correct size.

Of course this does not mean a 60x40 image from an iPhone will not be pixelated but any "decent" size image will be fine. I have a 30x20 inch print from a file less than 1mb (taken on a 10 MP dslr and severely cropped) and it looks great.

Modern rendering and / or RIP software that the labs use to do this is extremely capable

If in doubt, get a test print or speak to the lab.
 
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