do you need a headset? that depends on what you prefer. some people like using higher quality headphones for better sound and a desk mic. some people like having a clip on mic. some people prefer using a headset. better mics mean you wont sound as distorted perhaps but as long as you are happy with what you have now then i say just stay with it. if you want something better then upgrade. your call.
i would definitely suggest getting some decent peripherals. while certainly you can use the ones you have getting a nice quality mechanical keyboard and a good mouse make a pretty big difference. while you wont be instantly better in games they can be more comfortable to use and give you a few options you may not have right now.
a mechanical keyboard can increase your typing speed and accuracy. its also not gummy feeling like some rubber dome keyboards.
a gaming mouse can have greater accuracy, give you extra buttons, include keymapping software and some include dpi switches in case that matters to you.
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what do i personally use? (click the link in my sig if you want a photo)
razer deathadder 3.5g
wasd keyboard (currently dead after a spill... using a temp right now... probably going back to my ibm m's as they are great)
large mousepad (i got it for $10 at a show)
ath-m50s studio headphones (not shown)
i dont use a mic often but when i do its a desktop microphone.
just pick and use what works best for you and your budget.