Yes everything will work just fine. Free to air TV here is crap, so you will likely get cable and the set top boxes have a wide variety of outputs (RF, HDMI, Components etc.)
DL is right, your landlord is very unlikely to reduce the rent for bringing your own. You might even find yourself being charged for removal and storage!! In my experience, I found most places (condo's) come partially furnished - which means whitegoods only.
However if you aren't paying for your own relo, think carefully about it. Quality brands such as Fisher & Paykel that you can buy relatively cheap in Australia are premium appliances in Singapore. Unless you get a brand new
condo, you are likely to be furnished with cheap, no featured, well worn whitegoods that will chew through your power bill.
Also, who knows how hygenic the last tenant was? I certainly wouldn't wan to to wash my clothes in the remnanats of some famlilies nappies or dog blanket!
Another thing to be mindful of is that the rental market is negotiable. Things you might have felt uncomfortable asking for back home are kosher in Singapore. Its perfectly fine for example to ask for a new coat of paint - something I would never have done for in Oz. How much are you willing to spend on a condo? If its a fair amount, then you would be in a good position to negotiate with the landlord for brand new applicances anyway.
We didn't bring our washing machine and regret it...