No
The rules
The main reason for this is that the Tentacle Rod makes melee attacks, while Divine Smite requires melee weapon attacks.
Melee attacks (PHB p. 195)
Used in hand-to-hand combat, a melee attack allows you to attack a foe within your reach.
A melee attack typically uses a handheld weapon such as a sword, a warhammer, or an axe.
A typical monster makes a melee attack when it strikes with its claws, horns, teeth, tentacles, or other body part.
A few spells also involve making a melee attack.
So melee attacks are considered melee weapon attacks only when made with a handheld weapon, otherwise they are plain melee attacks (coincidentally, tentacles are actually given as an example of this case).
The difference between the different types of attacks is very well explained by Dale M in his answer to a different question, but to reiterate:
There are 2 axes of attacks: every attack is "weapon", "spell" or "one of the specifically listed things that are neither (e.g. unarmed, grapple)" and "melee" or "ranged".
Additional musings
Tentacle rod
While holding the rod, you can use an action to direct each tentacle to attack a creature you can see within 15 feet of you.
Each tentacle makes a melee attack roll [...]
Given that the attack is made by the tentacle itself, while the PC uses his/her action to use (command) the rod (which isn't a function of the Use an object action - DMG p. 141) and not take the Attack action, this could also be argued that disqualifies the attacks for the Divine Smite requirement, as you have to be the one making the attacks:
Divine Smite
Starting at 2nd level, when you hit a creature with a melee weapon attack[...]
Jeremy Crawford unofficially tweeted that this works for wands, though the rules do not mention it.
Since the only hint (apart from their names) that magical rods, wands and staves can be used as foci are their tags rod, wand and staff that classify them as such, then we look at their tags descriptions at pages 139-140 of DMG but as you previously said, there is no explicit information about their use as arcane foci.
However, according to Jeremy Crawford, any wand can be used as an arcane focus. I can't see a reason why it wouldn't also apply for rods and staves since they usually are also foci but none of them had it written on their descriptions.
@JeremyECrawford Can Wand of the War mage be an arcane focus? Doesn't specify, and while wands can be foci, it's not clear that ANY wand can
JC: Any wand can be used as an arcane focus.
Jeremy Crawford tweets are no longer official rulings; only those in the Sage Advice Compendium are. However, the SAC does not provide satisfactory answer on this subject, so Crawford's tweet does indicate the designer's opinion on the matter, for what is worth.
The descriptions of DMG magic items are not exhaustive.
Marq commented:
An arcane focus is defined as "a special item ... designed to channel the power of arcane spells". Magic items of the "rod" category are "typically made of metal, wood, or bone ... about 2 or 3 feet long, 1 inch thick, and 2 to 5 pounds." Not all mundane rods of those dimensions are arcane foci (some may be, for example, cart axles, or crowbars); thus not all magic rods are. In contrast, all swords are weapons, by definition; it's not a special subcategory of swords that are made for attacking
First, I wouldn't call magic rods and staves something mundane; they are magical.
Second, the DMG magic items descriptions (p139-140) don't describe their every function, they do not specify their most basic functions since magic wands have nothing written there that they can be used as foci as weapons doesn't say that they can be used to attack. The same thing about rods and staves that doesn't explicitly say about their use as foci. RAW is the same to say that I can use a normal sword to cut someone but I can't use a Dancing Sword to do the same because isn't written that I can unless I make it fly while in my hand it doesn't have the capability to injure anyone. That would be unreasonable.
PHB also doesn't help.
There are no non-foci rods or wands described in the PHB. Would all rods and wands in existence be foci? That would also be unreasonable since it clearly says they are specially prepared to be used as such. If you imply that someone could make a non-foci rod with a log to support a broken table or a non-magical wand with silver and jade for a teacher to point at the blackboard because it's reasonable but isn't written in the book, then someone can make an ornamental sword that isn't a weapon and also isn't written there. Just because isn't written in the book doesn't mean it can't be done.
Marq said in his answer:
Magic items that can be used as a spellcasting focus, such as a hat of wizardry, say so explicitly:
This antiquated, cone-shaped hat is adorned with gold crescent moons and stars. While you are wearing it, you gain the following benefits:
- You can use the hat as a spellcasting focus for your wizard spells.
Unlike wands, staves and rods, hats are not usually arcane foci. If a specific item that usually doesn't have such function now works as such, obviously it need to be stated in its description. That's the reason the hat of wizardry clearly have it noted there. Since wands, rods and staves are meant to be used as foci as much as weapons are meant to be used to attack, they didn't need to have it noted in their entries; it's their basic function, you just assume they do what they are supposed to do.
Best Answer
Multi-attack is an action, so tentacle rod would not prevent multi-attack.
Players get multiattack too (of sorts), not just monsters. The Fighter gets Extra Attack at 5th level. The class feature description states:
Multiattack works the same way for monsters as Extra Attack does for players (in terms of action economy). It only takes one action and is not split up between the action and bonus action. That's why the game is very easy to run. You get very general rules that apply to many instances of the game. Only if there is an explicitly stated exception to a rule do you break it.
Regarding example 2 from your question:
Monsters and player characters are designed differently. A monster is generally stronger than a PC in certain ways, but PCs have the advantage of not being run by one person (among other benefits). That being said, monsters can have certain attack options that PCs cannot.
The Tentacle Rod does have some cool effects. If the monster has a use for its bonus action like casting healing word or using spiritual weapon, then it could not do those actions on the same turn it attacks, because they use bonus actions. In the same way, an evil wizard could not use counterspell because it is a reaction. However, Multiattack counts as one action and is not stopped or split up.