could the easterling breach erebor?

[Zigûr]

Originally Posted by tom the eldest

Isnt iron hills dont have a sizable number of dwarves anymore since the refounding of erebor?im pretty sure that it has become obscure since erebor was refounded.

There's no definitive statement either way, but as I say given that we have no real mention of the Iron Hills playing any role in the affairs of the last years of the Third Age it seems not unlikely that they were at least depopulated after the reclamation of Erebor. It might be reasonable to imagine that the Dwarves may have kept their mining operations running there, as they did in the Blue Mountains, but it doesn't seem especially likely that they kept a substantial settlement there after Dáin became King Under the Mountain, certainly not enough that could relieve a siege. The Dwarves' numbers were dwindling by that time at any rate.

Originally Posted by tom the eldest

as for catapult,they could get it from sauron,and that catapult could add some firepower,because its able to throw high enough to bypass minas tirith's first wall,which is very high,while at the same time remain out of range from gondorian machine in the wall.

As I said. Erebor is a mountain, however, not just a walled city, so it's debatable as to how effective a catapult would be regardless of its trajectory. There'd be nothing to fling missiles over and one could hardly expect that they could beat the walls of the mountain down. The Easterlings "could not take the Gate" so it seems that beyond that there were few if any major exploitable points on the exterior of the Mountain.
Again, they could have received siege engines from Sauron, but we have no proof that they did, and given that Sauron's engines seem to have primarily been manufactured in Mordor it's unconvincing that they had access to them. This Easterling army seems to have derived from northern Rhûn, distantly from Mordor, and they did not receive aid from Dol Guldur because of Thranduil (assuming Dol Guldur, in the middle of the forest, even had any storage of such artillery), so their access to advanced engines seems not especially likely to me. If they couldn't take the gate, whatever artillery they had must not have been very effective, or the defences were simply too strong.
The Easterlings' main aim seems to have been to capture the gate, presumably by storming it with Men and perhaps with rams and ladders, but their main recourse like so many ancient sieges would probably be to starve the enemy out. I think had the Easterlings not had cause to waver, the siege would have been protracted and bloody with no clear outcome unless either side could bring reinforcements to their succour.

[Tuor in Gondolin]

A siege of an entire mountain (Erebor) brings to mind the 8-month Crusader siege of Antioch during the First Crusade. There werne't enough men to meffectively close off even the city, a mountain, with dwarf engineering technology able to send out scouting groups, raiding/harassing parties etc. at their own initiative would be highly difficult (at least, without a dragon patrolling around Erebor ). There would presumably be enormous food and storage prepared (again, not unlike castle preparations in the Middle east with large water retaining stone pits). And even Thorin and Company were prepared to put up resistance on a hastily build wall against the combined elf/man army.