Actually, it's the smaller spiders you need to be aware of, i.e. the redbacks. The larger ones are actually quite friendly and non-venomous. Hooray. Also watch out for the snakes as we have 9 of the 10 most venomous land snakes in the world (not the Black Mamba though, thank god, that thing is in a league of its own when it comes to snake insanity - if you catch it then release it somewhere else, it will turn around and look for you, then chase you. Mortality rate for an untreated bite is 100%. You can keep that one, Africa).
As for staying somewhere near Uluru - I have heard it's better to do a day trip and stay a bit further away. The tours I have also heard are very dodgy, so it's better to do it yourself. I'd give yourself a day to visit Uluru (try to see it at sunset, apparently it's amazing - haven't been, so I can't say for sure though), and a day to visit the Olgas nearby.