How to stop background from getting dulled

If you are working on web graphics make sure your Transparancy blend options are set to RGB. (Starting with a Web or screen preset should set that) This should change so that you don't "Dull graphics".

Whenever transparency is used InDesign is forced to siulate transparency flattening and if this is sett to CMYK you are making InDesign simulate print (CMYK) colours, and it is this that you are reacting to. If the end is Print it is infact giving you a more realistic expectation. and if you have proof colours enabled you do not see colour shift since you are already in the "CMYK" view mode. (I see the Adobe Help does not explain this where it explains blend modes)

I agree that it's not the best for this kind of work, but you can use InDesign if you want, but you need to Change your Transparency Blend Space (under Edit) to RGB. Oherwise, as you've seen, any transparent object, like your PNG will force the the entire page to convert to CMYK as well, hence the duller blue.

To start with, InDesign is not really a good tool to do small web graphics. It can, but getting a quality image without resolution issues or, say, color shifts from what is primarily a print/publication tool takes some jumping through hoops.

Use Photoshop instead. Really.