A while ago Google created a 'mobile friendly' test and now advises that websites should be built in such a way that they 'pass' this test.
Yes, it is important now because it may affect future Google searches. Despite this, alongside having well written code, genuinely reflecting the business it is representing and being visually appealing to attract new clients or customers are still the most important criteria for successful web design.
Responsive Web Design is the process of building a website that is able to adapt to a number of different sizes of screens. It will 'respond' to the size of screen it is being viewed on. Some elements of the website may be coded to only appear on larger screens, eg. large images, or they may be coded to 'respond' to the size of the screen, ie. those large images will reduce in size when viewed on a smaller screen.
No, even though many web designers do offer this, because the designers rarely have the expertise to ensure that having two websites for the same business doesn't confuse the search engines.
It has been reported that websites that are mobile friendly will do better in mobile searches but many studies have been done that show that right now there is no difference between the success of a website that passes the Google mobile test and one that does not. The assumption is that eventually mobile friendly websites will do better in search engines when searching using a mobile device.
Yes, all Webgloss's new websites are designed so that they pass the Google 'mobile friendly' test and many of our older (pre 'mobile friendly test' era) websites are being redesigned. We use 'responsive' web design.