Q: Why did my BTC balance go from $10.39 to $7.14 straight away after I clicked Exchange?
Is that the difference in money going to shapeshift.io for services rendered (if I went ahead)?
Q: For larger amounts from BTC to ETH is it cheaper to use a bitrex or poloniex for exchanges from BTC to ETH rather than Exodus?
The difference in money is money going to shapeshift.io (small part) and transferfee (biggest part). Transferring bitcoins is expansive unfortunately.
Bittrex and Poloniex have sometimes better rates then Shapeshift, but not always. Transferring bitcoins to those exchanges will be expansive also, and they are taking part of the exchanged amount (0.15 - 0.25%) also.
Big advantage of Exodus using Shapeshift instead of exchanging it yourself (Yes, you can use Shapeshift outside Exodus as well!) is the service of Exodus.
When something goes wrong you have to contact the very good responsive Exodus team for support and they will help you out. This is much faster then the support of Shapeshift, especially when there is a high load on support. BTW support of Shapeshift is much faster then the other exchanges is my experience.
So for comparison: Exodus support is much faster then Shapeshift support, which is much faster then Poloniex/Bittrex support. The latter two are bad in handling support tickets, it can take weeks before they answer, while Exodus is hours. And yes, Exodus corrects the error before contacting Shapeshift so in those cases they lend you the money (!). Fortunately those mistakes aren't very common.
