Ghosts of Onyx actually goes into quite a bit of detail about the Spartan-III programme, so if you haven't already done so, it's definitely worth reading if you want to find out more about the Spartan-IIIs. Admittedly it's been a while since I last opened it, so a lot of this could be wrong, but from what I remember it's never actually made clear whether or not Spartan-IIIs are as strong as Spartan-IIs: Gamma company in particular is very uncertain, due to the fact that they received extra enhancements which meant they could go beyond the human body's normal limits.
As for whether Spartan-IIIs could use Mjolnir - yes, there are at least six documented instances, as the members of Noble Team used Mark V Mjolnir armour during the fall of Reach, and Thom-A293 wore it during a previous mission. There's concept art for Rosenda-A344's armour as well, but as she's never actually seen wearing it in any Halo media we can't be sure that she actually was capable of wearing it. However, it isn't clear whether or not those Spartan-IIIs seen wearing Mjolnir armour received any additional augments or not in order to do so, as prior to Reach, Spartan-IIIs were only ever seen using weaker, SPI armour.
Notably, although Noble Team seems to have filled a role somewhat similar to the Headhunters, another Spartan-III special ops team, the latter only used an advanced model of SPI armour, even though Mjolnir would seemingly have suited their role better: however, it's possible that the Headhunters were, like most Spartan-IIIs, just considered expendable, and so weren't equipped with better armour for simple budgetary reasons. Eventually, a lot of surviving Spartan-IIIs were adopted into the Spartan-IV programme, in which everyone wears Mjolnir armour - but again, it's never made clear whether this is because the Spartan-IVs received the same enhancements as the IIIs, better enhancements which were retroactively granted to the IIIs, or simply because the new generation of Mjolnir armour needed less augmentation to safely operate than its predecessors. In any case, ultimately at least some Spartan-IIIs were capable of wearing Mjolnir, so in broad terms, the answer is yes.
In terms of raw strength, reactions, and toughness, it's still unclear, but there are a few examples within the games. At the start of Reach, Noble Six is pinned down by a Covenant Elite, and although capable of punching the Elite, still needs help from the rest of the team to throw it off entirely: by contrast, during the second encounter with the Promethean Knights on Requiem in Halo 4, the Master Chief is able to free himself with a single kick. Unfortunately, as usual there are a few things muddying the waters here - firstly, the Chief is wearing a significantly more advanced version of Mjolnir armour than Six was, meaning it's difficult to tell whether that extra strength was simply due to his suit, and it's also not clear exactly how much a Promethean Knight weighs in relation to a Covenant Elite - it's possible that they're actually not very heavy, as there are a few hints that they, like everything on Requiem, are formed purely from hard light rather than actual materials.
Another set of examples can be found in the various falls from space - Six suffers moderate injuries after his fall from the Ardent Prayer in the aftermath of Operation Uppercut, while equipped with a specialised re-entry pack: however, the Chief is back on his feet, apparently unharmed, within minutes of a similar fall from space with nothing but a large sheet of Forerunner metal for protection. Again, it isn't clear whether this is due to the more advanced Mark VI Mjolnir suit he was wearing, which as we've seen in Halo 4, may have had some sort of thrusters built into the armour itself to help slow him down, or simply due to the game mechanics not including a visible health bar.
What is certain is that Spartan-IIIs were at least as capable of the same levels of combat effectiveness as Spartan-IIs: Noble Six is specifically noted to be the only Spartan, from any programme prior to the fall of Reach, to hold the same rating as the Master Chief, and while he may have been unusually effective it at least proves that Spartan-IIIs could match Spartan-IIs, at least in combat abilities.