In my experience, you can fly at any reasonable speed (not above 400-ish) in controlled airspace as long as you’re complying with IFATC instructions and not causing trouble. Always keep your eyes peeled for potential conflict using the mini map and by looking out the cockpit. If you detect a potential hazard, don’t wait for IFATC to tell you, act immediately to eliminate or at least minimize the hazard. A fighter is manoeuvreble and you will have lots of surplus power, giving you plenty of options.
If you want to fly fast (500 knots+) then you should not do so in IFATC airspace. If you do, you run the risk of being ghosted without warning.
If nobody is around then nobody cares. If in doubt, don’t do it.
I will often come barrelling in at 400+ knots before reducing to something around 350-ish. I reduce my speed further when requested or when I feel it is necessary.
As mentioned above, entering IFATC airspace at very high speed renders you liable to ghosting without warning.
I have never been ghosted by IFATC when doing any of these things. Always exercise your own good judgement when flying, and follow all IFATC instructions promptly if able.