You could use the Menu attributes module to add classes to each module (such as top-left, bottom-right, etc) and style based on those. This will be a lot more robust than using ids derived from the menu link's mid.
Next, to get the links to the corners of the screen, you will need to give them position: absolute or position: fixed. Bear in mind that that will allow you to position them relative to their first positioned (not static) ancestor element. This means that you'll have to make sure that none of the containers around the menu links have a css position of anything other than "static".
Finally you may need to add "min-height: 100%" to the html and body tags in order to make sure that the bottom-left/right links do get down to the bottom of the page.
These changes might break other styling on your page, which would probably mean that one of the containers around your menu links needs to have a position other than static. If you can't work around this then you'll need to move your menu outside of that container by (for example) overriding the relevant templates such as page.tpl.php, or if all else fails, you could use javascript to move the menu to another part of the DOM where it's styling won't effect other elements on the page. However, you should only do that if you're happy that non-javascript users will still get a sufficiently usable experience without the links being at the edge of the page.