From what I recall from my childhood, physical typewriters worked slightly differently: the accent keys were non-advancing ("dead") keys. You pressed the "acute" key followed by the "e" key for an é, for instance. If you wanted a bare accent, you pressed the accent key followed by the space bar.
(The typewriters I recall also didn't have a 0 or 1 key, you used uppercase O or I for these numbers.)
Yes. I consider it a mistake of Unicode that combining characters follow rather than precede the base character. If they preceded, most dead keys could simply generate the appropriate combining character, rather than requiring complicated input method support. (And finding the end of a sequence of multiple combining character wouldn't require lookahead.)
The Unicode way makes sorting easier. Your way would require special knowledge about the characters to know that ä should sort directly after a, rather than directly before ë.
I believe that depended on the manufacturer and country convention. Most US keyboards didn't have an accent character. For example, here's one from the 1950s:
For acute or umlaut you could use a + backspace + ' or u + backspace + " (or the opposite order). For grave or circumflex, I don't think there was a solution. Write it in by hand?