Is it possible for a self-replicating cell (that means it has a membrane or some form of protection, it has "string-like instruction molecules" and it has molecules to carry out these instructions at the minimum) to be able to exist and function properly?

Yes! Apart from the radically-different-physics cellular automaton examples referenced in other answers, DNA is already pretty much one-dimensional, and its copying process need not require more than 2 dimensions. A 2D linear genetic molecule can replicate, or permit itself to be read, essentially identically to a 3D linear genetic molecule--by unzipping down the middle, and accumulating complementary monomers on the exposed edges of each half.

If it can (I think it would be able to), would this cell be able to evolve?

Evolution only requires replication with variation. The self-replicating cellular automata described in other answers do not permit variation during replication (being completely deterministic, they make no mistakes--and if they did make mistakes, they would almost certainly be "fatal"). A 2D DNA-analog, however, would be just as susceptible to mutation as 3D DNA, and there is equally no barrier to 2D sex at the cellular level.

If this cell can evolve, is multicellular life a possibility? (e.g. 2-eyed creatures moving around with flippers and eating bits of edible material with mandibles or something like that)

Yes. The canonical example of such life is the Ardeans from The Planiverse. Their biology is heavily based on zipper-mechanisms to permit the passage of materials between the exterior and interior (i.e., eating and expelling waste) and between different sections of their bodies (pushing food along the alimentary canal, pumping blood). As a result, most Ardean creatures are actually two separate halves "glued" together most of the time (there are two exceptions in the book, the intelligent Nsana and their closest relatives, which eject waste through their mouths). Zipper-lock mechanisms are found all the way down to the cellular level, as part of a multi-layered cell membrane. Personally, however, I find that construction to be implausible at the cellular level; vesicle-based transport, combined with a lot more multi-nucleated "supercells" sharing a common cytoplasm than we are used to in our 3D biology, seem like much better solutions to the problem of transporting metabolic materials at the cellular level. Vesicle engulfment and expulsion may also be a decent transport strategy for macro-scale multicellular organisms, but velcro/zipper arrangements do make more sense at that scale.

The other major issue that would need to be solved by complex animal life is signal crossing. Nerve networks just can't be made fully planar. Fortunately, however, there is a clever arrangement of 3 XOR gates that allows signal cross-over in only two dimensions, which should not be hard for biology to discover and implement in neurons.

There is, however, another avenue to solving the crossover/transport problems, which may be exploited in parallel with those already mentioned, or possibly instead of: mutually-non-excluding matter. Solid materials in our universe cannot pass through each other fundamentally because of the Pauli exclusion principle; electron clouds bounce off each other not because they repel each other electrically (atoms are neutral, after all), but because multiple fermions of the same type can't occupy the same space at the same time. Change the type of particle, however, and exclusion no longer applies! (This is part of why, for example, muon-catalyzed fusion works--not only is muonium smaller than regular hydrogen, but a muon cloud can pass right through an electron cloud with no resistance.)

So, consider (for example) a 2D universe with protons, neutrons, electrons... and magnetic monopole parallels for all of those. Each type of matter would still be able interact and form (weak) bonds with the other, such that you could form hybrid organisms composed of a mix of each type of matter (thus it is useful to eat both types of matter), but each type of matter would also be fully capable of passing right through the other, so it can be injested through special mono-material ports made of the complementary matter type. And of course, there are innumerable other ways you could modify the matter of your 2D world to make life easier--electromagnetic symmetry is just one off-the-cuff example.