Do airlines continue operating in the vicinity of a hurricane?
The cross-wind limit is 35-40 knots for most aircraft with gusts allowed to perhaps 45 or so. The head-wind limit is slightly higher, but not much.

Knot is nautical mile per hour. Nautical mile is 1.852 km, that is 1.15 mile (and for those who prefer metric, knot is just a bit over ½ m/s). So wind more than 40 miles/hour may interrupt airport operations and wind more than 50 miles/hour surely will. I don't think there is a specific limit for rain; it is generally not a big problem if the runways are in good condition.

Problem is that good local forecasts are possible for at most around 24 hours. So it's not possible to tell whether it is likely there will be disruption of operations the day after tomorrow.

For one day ahead, the airport forecasts are publicly available on the internet. Search for TAF ("terminal aerodrome forecast") with the ICAO (the 4-letter one) code of the airport. They are published in strange abbreviations, but there are sites that will happily decode it for you.