Sir Dinkmeister- first of all let's have a look at your iron. The typical iron usually draws around 1200 watts. At 120 volts this looks a lot like 10 amps. Typically a circuit breaker for a receptacle is rated at 15 amps using 14/2. Code allows you to load up a breaker to 80% of its max so with the iron you're at 66% which leaves only 12% (2 amps.) However, if you're not popping a breaker I wouldn't worry about separating your iron from everything else as your iron may have a lower power rating.
Now, as for the flickering lights, TYPICALLY the lighting is put on a 20 amp breaker using 12/2 or 3. So in this case putting the iron on a separate breaker wouldn't do much good in solving that problem.
As for your computer, it uses very little in the way of actual power. The main thing you want to protect your computer from is voltage spikes. Your iron isn't causing a spike in the voltage. If anything it may cause a temporary drop in the voltage so in my opinion, putting your computer on a separate circuit isn't necessary because as I stated earlier, your lighting is on a separate circuit and it still flickers. That tells me that whatever the iron is doing, it's feeding back clear through to the main bus that you plug your breakers into.
Take my opinion with a grain of salt as I don't claim that the buck stops here. This is just what I know from my own experience. My iron does the exact same thing but I'm not worried about it in the least! Have a good one Dink. Best, Elder.
PS- If anyone knows something I don't, feel free to put in your 2 cents. :)
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