The concept of an expensive engagement ring is based on the realities of a different era, as is asking her dad for permission.
At a time when women couldn't have bank accounts, their jewellery was their emergency savings. The engagement ring in particular was a way for a man to prove that he could financially provide for her, and for her to show to the world that she made a good financial match. This is why engagement rings have historically been very expensive status symbols.
Today, the engagement ring is about love, not status. So forget about what you "should" do and show her that you love her, by thinking about what she would appreciate most.
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I wouldn't ask her dad unless she is very traditional and extremely close to her dad i.e. only do this if you believe that she would like that and that he will say yes (and yes, it is very patriarchal).
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What do you think she would value more - having you propose with a ring, or going shopping for a ring together? She will have to wear it, so making sure that she loves it is more important than the price.
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Stick to what you can realistically afford, bearing in mind that weddings and honeymoons are expensive too. Might she prefer a less expensive ring and a nicer honeymoon?
3a) Definitely don't overspend on a natural diamond. The value of natural diamonds in typical engagement ring sizes (< 1 carat) have fallen dramatically in the last 10-15 years, you can see a fun chart here: https://www.pricescope.com/diamond-prices/diamond-prices-chart/ (Ask me how I know π)
Above all, remember that this is meant to be a fun and exciting time for both of you, so forget about what others expect and focus on what would make you the happiest in the long term.