Lets answer the Qs all in a shot.

You can store you hand carry in the Silver Kris Lounge, where you can carry the key. However, the size of the lockers in SK lounge is unknown to me. (Never been there)

2nd alternative is to store the stuff at the Left luggage area INSIDE Transit, just behind the Central Duty Free Liqour Shop. It is constantly manned but closes at 2230hrs, so you need to take out the items before that time.

3rd alternative is to store it at the Left Luggage at the Departure Hall near Check In Row 1. It is operational 24hrs.

Points to note:
1)Pls remember to take your connecting boarding passes together with your passports for clearance when going out of immigration, otherwise it would be a bit of a hassle.

2)You checked in bags should go straight thru' to your final destination, so even if you go to baggage claim, you may not even see your bags, unless you requested only to check to SIN.

Will,
From your flights, it's best you store your bags at the left luggage outside (3rd alternative above).

J777