Thanks Ronpat for your answer although I'm not sure I deserve the tone.
I'm quite new to mobile development so I'm just trying to understand why I'm getting so many errors. The website is to be viewed on smartphones and I actually thought that smartphone browsers were on average more evolved that some of the old desktop browsers.
We're getting a really good grade on W3C mobileOK Checker http://validator.w3.org/mobile/
So I'm just surprised that so many properties such as border-radius or even inline-block are seen as errors by the W3C CSS Validator in "mobile" mode.
I gave the link to that service in my question, so that answers yours about how I test my pages (http://jigsaw.w3.org/css-validator/)
To tell you the truth I'm working as a project manager on this project which was already built before I joined the company. I used to be a front-end dev so I understand HTML/CSS. The client has asked for improvements they're entitled to but I'm running some tests to see what we could improve and optimize.
I didn't expect to have to "optimize" or even remove things like border-radius.
I would think that, just like on desktop browsers, if a browser doesn't support border-radius it would just render square corners. Why generate an error? I'm trying to make sense of this, that's all.
I'm wondering what to say to the client because I don't think they'd want to give up border-radius or box-shadow.