The chances of comming across a cockroach is very high. But they are the least of your worries.
As for spiders, the big ones are harmless, infact I have an 8cm huntsman spider that lives in my bedroom, I don't mind, he keeps the bug population low.
The spiders to worry about are the red back spider & the funnel web spider:
http://www.nikkish.net/jeff/blogpics/053005_redback.jpg
http://www.calacademy.org/exhibits/venoms/images/00_final/sydney_funnel_web_spider.jpg
both of these spiders are quite small.
The red back lives dark places & normaly won't bother you unless you litteraly put your hand right on top of it. If you do, you'll know about it. Very painful.
The funnel web is slightly larger & likes to live in shoes, so check your shoes before putting them on, they have highly toxic venom & can kill you. They are only found around the Sydney area tho.
You'll find the mosquitio's more annoying than the cockroaches so bring plenty of bug spray (well, you can buy it here).
As for living in Sydney, you shouldn't come across anything else to worry about. Most snakes won't come into built up area's.
Oh actualy, around the Sydney coast line there are blue ringed octopus, they are deadly as well
http://idealismodebuteco.files.wordpress.com/2008/03/blue_ringed_octopus_465x310.jpg
Their not found at the beach, so swimming is fine :) Just don't stick your hand under ledges & in cracks in the rockpools, on the rocky points.
With all these animals, the basic rule is 'you leave it alone, it'll leave you alone'.
Overall, don't be put off. The chances of comming across one of these is very low. Also, don't be scared to head out of Sydney & checkout the surrounding country side (Blue Mountains are close by)