>Would a ship with a nice mixture of 76.2mm and 155mm guns... have everything it needs to defend against most aerials and surface threats?
Theoretically, yes. Assuming the 155mm has everything that the 127mm/5in guns have at minimum and the developments of land-based 155mm guns.
>Or are the smaller guns unnecessary and it would be better to put some autocannons on it?
The balance of fire rate and effect benefits the 76.2mm guns in this comparison, while the effect of range benefits the 155mm (assuming AA shells such as HVP or hypothetical large DART).
You would want smaller cannon in the 20mm to 57mm range regardless.
Of course, present laser weapons (that have at least reached operational testing stage) invalidate part of this.
>What would be the ˝minimal armament˝ required for this job?
Surprisingly little. A modern late-WW2 design CL would qualify for a Gun-based Battlecruiser anymore.
Understanding that Guns themselves are not actually required for Battleship/Battlecruiser status (they are terms defined by roles, not weapons), a NATO aligned CC could simply carry VLS as its primary weapon and get by, so long as it was capable of performing its role of being the aggressor. As I suggested before, if the Kirovs had been designed with heavy armor, they would have been proper Missile Battlecruisers. As they are, they properly are Large Missile Cruisers, modern missile-equivalents to the Alaska-class.
>Especially in terms of big guns.
Limiting to big guns, you don't actually even need scramjets. Those are the upper bounds of what is nice to have, but they are not actually required.
With modern metallurgy, it's possible to create big bore guns that can hold up to chamber pressures multiple times higher than what the old Battleship guns could bear, so you can use some rather 'hot' charges and, with guidance systems, get drastically increased ranges. That was the concept behind the EX-175 charge and, when mated with the HVP, gave demonstrated ranges exceeding 50nm (89km) out of a 5in gun.
All of this being said, I would personally pursue naval versions of the US Army's SLRC. Which is, according to (unreliable) scuttlebutt, a roughly 12in cannon. Roughly 6-8 of them per ship, probably in two-gun mounts. It would give the ship a ~1000mi range, which far exceeds the strike range of CV based fighters-bombers.
>with a payload that is about equal to what a Tomahawk carries
Ironically, to have drastic effect on a battlefield would only require a shell to have as little as a 25lb bursting charge. Essentially deliver an 8in shell 200 miles inland within a usable time frame and the land-based elements would love you. At present, it takes CVs over 2 hours to deliver even that, by which time any worthwhile target would have gotten out of dodge.
Obviously, we're talking minimums here. Although the 8in effect is the preferred result according to ground troops
>Are the advancements in tank armour useful here, or is it still better to just slap a thick steel plate to the ship?
It's far more economical to just slap thicker steel on the hull. You could use composite armor (and in fact, historical battleships such as the Iowa and even Yamato actually did this to some degree - the physics behind composite armor was already well understood, it's not something Chobham came up with), synthetics such as kevlar (the US CVNs do this), or all the way up to CNT-based composites, but if you're talking cost effectiveness, raw homogeneous steel thickness is hard to beat.
>What would be its job in this day and age? Hunting down enemy warships until it comes across something it can't kill?
Essentially, that and Naval Strike missions, it could (and probably would) end up serving as a glorified bodyguard for assets such as a CV/N these days, though. Literally the job of a Battleship, but details.
>On that note, I assume a modern battleship would be a fast battleship by default. If that is so, then wouldn't the very first battleship division with CNT armour make all these battlecruisers obsolete (at least on paper)?
On paper, absolutely. The super-tech BB would essentially delete any of these Economy CCs that it comes across.
The difference paper and reality is, unlike historical BB to CC to CA/CL/whatever comparisons, there actually could be many of these Economy CCs for every singular super-tech BB, so while the BB is taking care of one (or 6) of the CCs, the rest of the CCs are threatening the BB's homeland.
And for even more cold water, there'd be even more SSBNs that would be far harder for the super-tech BB to respond to.
Obviously, lasers, but surely you get my point on the eggs and baskets issue.