Is it possible? Yes, the 1.5 gearbox will bolt right up (you'll need to use the 1.5 clutch too), though you'll face some difficulties with coolant lines from the head to the heater matrix, and you'll need to make up a blanking plate or fit a freeze plug where the MX5 thermostat was. You'll also need to find a set of 1.8 BA engine mounts...
Would the gearbox hold up to the power of turboing it? No
You might be alright for 150-175bhp but after that you'll be pushing it! The older 1.8 G series box is a bit stronger but even then, must past 200bhp you'll risk stripping 3rd gear or snapping the shafts and casing.Is it an easy swap? Relatively, if you know what you're doing - wiring will be the biggest headache if you haven't got a 1.8 BA donor.
Potential easy swaps: 1.8 BA - that's the full list, anything else will be hard work, and you'll need a running and driving donor car anyway... which leads to the question, why not just use that car?