Why can alcohol undergo elimination just by sulphuric acid?
In E1, the way I understand it is that the resultant carbocation's protons become reasonably acidic as the conjugate base (an alkene) is very stable. They only require a weak base to be removed, so practically anything can remove them. It looks to me as if the likely weak base candidates are the water produced from the leaving group, or even another alcohol in the reaction mixture.

Primary alcohols will be less susceptible to dehydration by this method, as they follow an E2 mechanism which progresses much more slowly in acidic conditions due to the absence of a strong base (which would usually act to 'shove' the leaving group off, a weaker base gives a weaker 'push').

I have seen a few schemes for E1 and E2 in which the hydrogen sulphate ion acts as the weak base; the usual assumption is that there is a weak base that is able to capture the proton, but we don't necessarily care what it is because anything will do. As long as there isn't another half decent nucleophile in the reaction mixture, elimination will likely prevail.

As you can see from this image I found in Clayden Organic Chemistry, the proton in the reaction mechanism is often drawn as just 'falling off', emphasising that any weak base is capable, and abstraction of the proton is very likely by a large number of species.