The catapults are plausible. The 'heat ray' setting the ships on fire is dubious. Modern tests that were done indicated that the conditions would have to be perfect and ships to be stationary at relatively short range for this to work.

Regarding the 'Archimedes claw', he didn't leave a design for the device so we don't know if he actually built one or if it was an embellishment, but we can look at what he did know in terms of mechanics. He understood levers so it was possible for him to build a claw with a counterweight. He understood buoyancy and so he would know that a relatively small force would be needed to capsize a vessel. On that basis, he had the mechanical knowledge to build such a device.

On the question of its effectiveness, modern testing suggested that it wouldn't be necessary to lift the entire ship out of the water, it was enough to drop the grapple next to it and drag it sideways. If the galleys had siege ladders deployed they would have been even more unstable. This is all not exactly evidence, but at least we know it was plausible and practical.

Here is what Livy says:

' The city wall ran over ground of varying altitude: in most places it was high and difficult of access, but here and there were level stretches which could be approached along low, flat ground; and along the whole extent of it Archimedes had moved into position the type of artillery which he thought suitable of the various sections. Marcellus, for his part, began the attack on the defences of Achradina, which, as I have said, run down to the water, from sixty quiquiremes. On most of the ships archers and slingers--and also light troops armed with a special javelin which anyone not used to it finds difficult to hurl back--made it almost impossible for any defender to stand on the wall without being hit. These troops kept their vessels somewhat away from the walls, as they needed range for their missiles; other quinquiremes were lashed together in pairs, one close alongside the other with the inner banks of oars removed; each pair was propelled, like a single vessel, by the outer banks of oars, and they were employed to carry towers, several storeys high, and other devices for breaching defences. This elaborate sea-borne attack Archimedes countered by moving into position on the walls pieces of artillery of varying size; at the ships off-shore he hurled stones of enormous weight, assailing those closer in with missiles which, though lighter, could for that reason be discharged more frequently. Then, to enable his own men to discharge their missiles at the enemy without danger to themselves, he made rows of loopholes, ranging from top to bottom of the wall and some eighteen inches wider, through which, themselves unseen, they could shoot at the enemy either with arrows or with smallish catapults. Some of the enemy ships came close in-shore, too close for the artillery to touch them; and these he dealt with by using a swing-beam and grapnel. The method was this: the swing-beam projected over the wall and an iron grapnel was attached to it on a heavy chain; the grapnel was lowered on to a vessel's bows, and the beam was then swung up, the other arm being brought to the ground by the shifting of a leaden weight; the result was to stand the ship, so to speak, on her tail, bows in air. Then the whole contraption was suddenly let go, and the ship, falling smash as it were from the wall into the water (to the great alarm of the crew), was more or less swamped even if it happened to come down on an even keel.

By these devices the attack from the sea was frustrated, and all available strength was diverted to an assault by land. Even there, however, every section of the defences had been equipped with various missile-throwing machines, all at the expense and by the forethought of Hiero over many years, aided by the unique engineering skill of Archimedes. Moreover, the nature of the ground helped the defence, in that the rocky cliff on which the foundations of the wall are built is mostly so steep that not only missiles hurled from catapults but even such as were allowed to roll down the cliff by their own weight, took deadly effect on the enemy below. The same reason rendered an approach to the walls difficult and gave but an insecure foothold to the attacker. Thus it was that at a council of war the decision was reached to abandon the assault, as all attempts were baffled, and to confine operations to a blockade by sea and land. '

So, a combination of geography, fortifications, and one man's ability to design war machines and add murder-holes. It doesn't seem implausible.