For example, if I live in the US and am trading TRYJPY, does an appreciation of the USD affect the returns of my TRYJPY trade?
If the USD appreciates (or depreciates) versus the JPY, then the P/L booked to your account (when your TRY/JPY trade is closed) will be affected.
At the time of closing your TRY/JPY trade, P/L (profit or loss) generated in JPY (the quote currency) is immediately converted into USD (your account currency).
• At the time your trade is closed, if the USD/JPY exchange rate is higher than you expected (when you planned your trade), then the P/L booked to your account in USD will be less than you expected.
On the other hand —
• At the time your trade is closed, if the USD/JPY exchange rate is lower than you expected (when you planned your trade), then the P/L booked to your account in USD will be more than you expected.
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The exchange rate has an effect on the trade surplus or deficit, which in turn affects the exchange rate, and so on. In general, however, a weaker domestic currency stimulates exports and makes imports more expensive. Conversely, a strong domestic currency hampers exports and makes imports cheaper.