I'm wondering if a complementary enhancement mode N&P channel MOSFET like this one http://micric.ru/data/documents/P2503NPG.pdf which is used as an inverter driver for the LCD display, could be replaced with two separate MOSFETs one N type one P type with similar characteristics? If not why not? What is unique in having those two transistors in the same package? Also if yes but with drawbacks(parasitic effects?), then I'm also wondering what could be added to the circuit to compensate for that.
Can you replace it with two discrete mosfets? Yes.
Should you? Much more difficult to answer. The pre-packaged duo has been matched so that the turn on and turn off times are similar for each. That SMD package would suffer less parasitic inductance than a discrete pair of mosfets. So far that has two things going way for it, as matching devices is difficult and not guaranteed to happen in the same batch of discrete transistors you buy (There are also fewer discrete mosfets that are made to be compliments to each other than there are BJT's, not quite sure why, but that's what I've seen.). The parasitic inductance really can't be overcome as it's a product of the packaging. Pretty much the only reason I'd ever not replace a part with it's identical or equivalent competitor's is if the product no longer existed. If you choose to go the discrete route, then please ask for help in selecting mosfets that meet your LCD's criteria in a new question, as that is a whole can of worms on it's own.