If we're talking 'bout responsive, you may use bootstrap (if not, start doing this).
When working with images, you should add the class img-responsive, this will modify the width of the image if necessary and the height will be auto, so if width decreases, height will decrease too.
You will always have an image that keeps the same % of its container and will never loose the aspect ratio.
There's no relation with SEO and image size declarations.
Page speed will be the same always, so if the image is 800 x 600 px, you'll load the full image, even if you declare it as 60 x 40 px.
You must think that, even using img-responsive, the max width and height of this image will be the real size of the image. So if we have a 800 x 600 px image, it will not enlarge it (because it'll become loosing quality).
So in 2016, it's recommendable to NOT declare height and width of an image. Instead use bootstrap's img-responsive class, other responsive framework class that gets the same result, or hand-made the proper jquery and css to reach the same.
Hope it helps!