In general, the tip to take the metro from the airport into town is a good idea, except when it is Friday. Being a Muslim country, Friday is a special day in the UAE and the metro only starts running at 1 pm on Fridays.
Of course, I got to Dubai on a Friday, but there should be buses, right? Yes, there are a bunch of them stopping in front of Terminal One. The problem is that you can't pay the driver but have to buy a ticket from the machine. With the metro station closed there is only one ticket machine at the whole huge airport. The ticket I wanted was the day pass and it was 16 dirham. Not only did the machine not give any change, it did not accept 'too' much money so I couldn't pay with two tens. I got some change, a tenner, a fiver plus one coin. Well, the coin slot did not work, so I couldn't get a ticket whatever I tried.
My original idea to walk into town did not seem like a good idea, it wasn't that hot but it was quite far and there were mostly freeways around the airport.
In the end, I met a friendly fellow who gave me a spare re-loadable ticket and helps me charge it with 15 dirhams. 14 is actually enough for unlimited rides during one day. It deducts fares for the first few rides and then stops at 1 dirham, all further rides are free.
To answer my other two questions: I could check my luggage all the way through to India with Emirates because I spent less than 24 hours in Dubai. But, yes there are left luggage facilities in Terminal 3.
I could have spent more time than just one day in Dubai, there are enough things to see and the climate is actually nice in February with a nice breeze.