As DM, you can do whatever you like, but I recommend following the guidance in Chapter 9 of the Dungeon Master's Guide.
The introduction to the Dungeon Master's Guide says (pg. 4):
The Dungeon Master (DM) is the creative force behind a D&D game. The DM creates a world for the other players to explore, and also creates and runs adventures that drive the ...
No, this won't work
All of the infusions that apply to weapons have the restriction:
Item: A simple or martial weapon...
A part of a sword isn't a sword and therefore isn't a viable item to have a weapon infusion placed on it.
Only one infusion per object
The rules in Tasha's Cauldron of Everything (where the Artificer is introduced) say on page 13 (emphasis mine):
You must touch each of the objects, and each of your infusions can be
in only one object at a time. Moreover, no object can bear more than
one of your infusions at a time.
It also goes on to say what happens if you ...
You can add weapons of your own design to the game, but be cautious. The existing weapons cover an enormous amount of design space, so making a new weapon that doesn't completely duplicate an existing weapon puts you in danger of making something that completely overshadows all existing weapons.
To me, there are two basic options: reskin or nudge.
Per the SRD:
To determine whether your target has cover from your ranged attack, choose a corner of your square. If any line from this corner to any corner of the target’s square passes through a square or border that blocks line of effect or provides cover, or through a square occupied by a creature, the target has cover (+4 to AC).
When making a ...
Yes, the DM can make whatever they would like for their game
There are two ways to do this: you can make your own weapon or you can modify a weapon.
Making a weapon is harder, but you have more control over what you want to come from it weather it be a flaming sword or never miss longbow. Making a weapon allows you to make a weapon however you want to fit ...
I'm going to go with yes.
The Halfling Sling Staff's description it
This staff ends in a Y-shaped split that cradles a sling.
Furthermore, we already have other items that 'count as' staves for attacking. Finally, the staff weapon is cost 0, meaning it's basically any stick you pick up from the ground that is long enough.
For these reasons, although it is ...
The DM determines if a beast is of the appropriate anatomy to wield a particular weapon.
There is no published guidance on what weapons can be used by what beasts. The Player's Handbook uses a phrase regarding mounts that is helpful here:
A willing creature that is at least one size larger than you and that has an appropriate anatomy can serve as a mount, ...
You can use Two-Handed weapons, including the Two-Hand trait, with HotA
You are able to wield your staff (or another weapon) in two hands while you cast Hand of the Apprentice.
Somatic requires that your hands are not bound.
You can use this component while holding something in your hand, but not if you are restrained or otherwise unable to gesture freely.
There is one such Feat... kind of
Virga May (Storm Hag) Changelings can choose Favorable Winds at 5th level.
You wrap a shell of favorable wind around a weapon you wield. On your next Strike with a ranged weapon before the start of your next turn, you don't take circumstance penalties on the attack roll due to wind. In addition, if that weapon ...
For those specific feats, technically a strike can be melee or ranged, so both feats work at range with this weapon.
With regards to the question in your title; 'Can a Halfling Sling Staff double as a Staff?'
Rules as Written (RAW), no you can't; you'd use it to make a ranged strike in melee, potentially causing an attack of opportunity, if a creature has ...
Hunt Prey, Kinda
I'm sure if you're already playing a ranger you've seen this, but in case you just missed it while theory crafting a build, Hunt Prey allows you to ignore the penalty for the second range increment, effectively already doubling your range:
You also ignore the penalty for making ranged attacks within your second range increment against the ...
In a normal game, yes, but not in Adventurer's League.
In a normal game, the GM is able to do basically whatever they want, as long as the players agree to it. However, Wizards of the Coast runs the Adventurer's League, an officially-supported shared campaign, and that includes additional rules for its GMs on what they are allowed to do and include in their ...
Very few beasts have hands, effectively, and can thus use weapons at all. Apes, maybe raccoons, octopodes, etc. So it's not really a big issue with a lot of choices. If the druid has some special weapon(s) made that they can use in animal form, that might work, but you're well into homebrew territory already there, so the DM will be able to decide as he ...
The damage die would be a d4 (RAW)
The spell has you throw the weapon, so you are not holding it all. You are definitely not holding it with two hands and ready to use it as required for the extra damage. Staves not held by two hands have a d4 damage die. (Explained in detail below.)
The damage die would be a d8 (RAI)
Paizo themselves have ignored the rules ...