The question I think is posted too many times, maybe a possible sticky?

As SIA and most probably its sister airline only operate departure lounges, you will have access to the contract SATS lounge iirc when you depart from Singapore, irrespective of the incoming class and airline.

Yes you will have to pick up your bags at baggage claim, I do believe in every terminal there are storage facilities, but can't remember the price

No T3 lounge sadly, read above.

Incoming at 0630, and outgoing at 1435, realistically you will be out by about 7am and will have to be back at the airport by 1.30pm, so whatever you do, you will only realistically have 3-4 hours in town, factoring in opening times and arrival/departure. You may want to just check out SIN, a pretty amazing airport, and the best I've been to.