Yes, when you have a large enough sample, you may have enough statistical power (precision) to show that even a small OR is statistically significantly different from 1 (even when the deviation from 1 is minimal).

"Thus, statistically the predictor has an effect on the outcome, but that this effect is very small and may be of little clinically or otherwise "practically" significant?"

Exactly.

The statistical significance is an indication that your data provides enough information about the effect to be able to distinguish the estimate (here: 0.9965) from the hypothesized value (here: 1.0). Since the estimate is lower than the hypothesized value, you can conclude that the (still unknown!) effect is also lower.

The confidence interval is the range of all hypothesized values for which the data would not be statistically significant. Hence, this is a range of effect values that are "not too incompatible" with your data. Values outside the interval are "too incompatible" with your data. You see that the (0.95-)confidence interval in your case does not include 1.0, which corresponds to your p value being < 0.05. So, whatever hypothetical value is not too incompatible with your data is lower than 1.0, but very close to 1.0. This may not include any value of clinical or practical relevance.

If OR is 1, you can't have statistical significance because the CI will always include the null value (1).

When it is very close to 1 but you still reach significance, it's probably that you have a very large sample size, enough to overcome the variation calculated based on the small effect. It may be a chance finding as well.

I always start with reporting the magnitude of the point estimate (OR itself). An OR=.99 for example, doesn't imply anything clinically meaningful at all.