Storage and space vary a fair bit from
property to property , but for $10K - 12K per month, you should be able to get somewhere that has what you need in terms of space. 2nd bedrooms often have closet space built in and if they are sharing a bunk bed, you should also fit a dresser. (I wouldn't bring much more kids furniture than that).
Typically, there is more space if you can find an older place (ie 10-15 years old) as the newer places are smaller.
If you're looking for partially furnished, almost all places have whitegoods, so I'd suggest that you don't need a washer/dryer, fridge or microwave. If you find a place with more furniture than you need, landlords will sometimes remove it. (I assume your company is providing a relocation agent if they are giving unlimited freight. Just have the agent ask the landlord).
The TV cabinet is really hard to say- it depends what the landlord already has in the place. TV areas are typically not built in, but the landlord might provide a cabinet. You could always leave it at home and as JR8 suggested, get an IKEA one if needed.
There are also loads of threads about schools on here if you are looking for somewhere for your kids. (Maybe check out Singapore American School - it has a good name and I note you're coming from the US).
All the best for the move!