Hello i think what your designer did here is ignoring the fact of legibility and readability and these 2 are main elements to design a logo that's why we create a logo guideline book so we can help the user to use the logo in multiple applications ( CMYK, RGB, BW , GREYSCALE, Printed, Digital etc...) and also in the guideline we include a color palette and the Min-Size of our logo.
On the other hand, the problem here seems to be a bit technical, while using illustrator. Your designer should always be sure that scale stroke & Effects is checked so the software by itself sets the values of all the effects, and when exporting the logo to png or jpg, he has to follow the right way : If exporting for web, he should be using save for web and devices option and to be sure to choose the right values of quality and colors. if exporting for printing we prefer that the file must be Using CMYK Format and Tif HR.
also there is one important element to be aware of. when you are using ur logo for the website make sure it is in RGB Format and not in CMYK Format, this can cause a bit of a problem when it comes to gradients and shadows.
at the end i also suggest that the lines around the logo has to be drawn not to be applied using the Bevel effect because this effect is knows to cause many problems.
As for the FAVICON let your designer create is in its real size, 16 x 16 or 32 x 32 and i advise u to use one color favicon cz its more legible.
If you want other advises on how to know if scale stroke and effects is checked of you would like me to suggest a favicon for you do not hesitate to ask