Any ship he wants, though the most mentioned are "the black ships", which seem to vary so wildly in design and origin that it boils back to, any ship they want...painted black, with no running lights active...to be more black.
Exterminatus tends to be carried out by a battlefleet, not a single cruiser
As far as I know there has been no mention of any single bomb powerful enough to destroy a planet. Even virus bombing, so evil its practically illegal require multiple drops and then further ignition weapons
A single dauntless class light cruiser has the potential to destroy a planet, not via full on kaboom levels of exterminatus, but with torpedoes, with a volley of the correct torpedo that one ship could in theory either burn the atmosphere away from the planet, or viral bomb it enough to be far too hazardous to live on, heck with melta torpedoes I reckon it would be possible for a light cruiser to crack a planet in two, if it has a week spot, sort of like firing a torpedo down a thermal exhaust on a space station
Inquisitors more often than not, commandeer Imperial Navy ships for their own use, or if going undercover, will commonly enlist a rogue trader/smuggler to ferry them around, some Inquisitors will even pay them, rather than comandeer them, to build a working relationship and be able to count on the captain and crew. The most prevalent examples of this being Eisenhorn and Ravenor.
Eisenhorn for a large portion of his career uses the sprint trader Essene, captained by Tobias Maxilla. Though it is less a case of Eisenhorn commandeering it by his authority, but more Tobias liking the thrill and excitement. Ravenor likewise, often employed the Hinterlight, a rogue trader vessel commanded by Cynthia Preest, regularly using her over his career. Both often used these vessels to remain incognito.
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