Browser testing no longer means attempting to make a website look the same in browsers of different capabilities or on devices with different size screens. It does mean ensuring that a person's experience of a design should be appropriate to the capabilities of a browser or device.
I test my work in current versions of major desktop browsers including those made by Apple (Safari), Microsoft (Internet Explorer) and Mozilla Firefox. I'll test to ensure users get an appropriate, possibly different, experience. I won't test in older browsers unless you specify otherwise. If you need an enhanced design for an older browser, I can update my quotation to include this.
Mobile browser testing
Testing popular small-screen devices is essential in ensuring that a person's experience of a design is appropriate to the capabilities of the device they're using. I test my work in Safari and Firefox on iOS 8, and in Firefox on Android 4.
I currently don't test Blackberry, Opera Mini/Mobile, Windows Phone or other mobile browsers. If you need me to test using these, I can update my quotation to include this.
Search Engine Optimisation
I cannot guarantee any improvement to a search engine ranking, nor can I promise to get a site higher up or to the 'top of Google', but I build every site in a way that is accessible to search engines in an effort to increase its chances.