The first flight out from MEL will be DXB one, so there will be staff around.
As you are going to the US, and as we know, the US are very strict, you might be able to do stuff towards OLCI, but most likely would need to have passport swiped/scanned/dipped by QFd staff at the T1 counters.
I would try to be at Tullamarine by 4.45am, lineup at the least.
As your 6.15am will be first QF flight out, to SYD, it will probably be busy.
And your flight, even tho you would have shown passport and have BP, it will still be a QFd flight, so you will still have to get from T3 at SYD to T1 via the QF bus, at gate 15, so, do it earlier.
If it was a purely QFd flight, then an hour with bag drop or 45mins at T1 with HLO should be enough.
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My flight departs MEL at 06:15, so I think the earliest flight of the day is only 15min earlier. But if I arrive in time for that one and there is space, I will switch, for the extra 15min in the F Lounge.
There is no access to the F lounge in MEL, as your flight to SYD, will be a QFd one, so will be departing from T1.
At SYD, yes, there is the F lounge there, I think it will be a bit early for the S&P squid, which I believe starts at lunch.
Have done them both when EK used to do MEL and SYD to AKL, with their F cabin.