What are the basic and upgraded Weapons called?
I'll start with this since i'll use these terms in my answers, in the Anime, weapons like Soul and Tsubaki are known as Death Weapons. After they collect the 100 souls they are refereed to as Death Scythes like Spirit and Justin, most likely a reference to how Lord Death is the Grim Reaper and in most works the Grim Reaper is portrayed to have a Scythe, these terms will be different depending on which version but for simplicity i'll use the anime terms.
It seems in the manga the 2 forms are Demon Weapon and Death scythes (デスサイズス desu saizusu) so I use the anime terms as Soul is refereed to as a demon scythe and using the anime terms avoid any confusion thinking Soul is somewhere in between.
What does completing this 100-soul ritual do for Weapon and Meister?
First things first, you ask about the effects of the ritual, to answer this you need to know what does collecting souls do for the Death Weapon. When a Death Weapon consumes a soul they get more powerful, the amount of power depends on thew soul they consume, Bellow are the list from the weakest to the strongest.
Kishin Eggs (Red with a forming Shell) are the weakest however since they can give birth to a Kishin the DWMA seeks these out.
Human souls (Normally Blue) are used to refer to the souls of humans who are innocent, ie. have done no wrong. They give quite a bit of power compared to Kishin Eggs however the DWMA looks down on this as you'll be murdering innocent people to get them and beginning down the road of becoming a Kishin.
Witch Souls (Normally Purple) give the most because of the magic Witches use however a witch is "normally" very hard to kill (exception may possible be Angela Leon due to her age)
The 99-Kishin/1-Witch Ritual is used to train both the Death Weapon and Meister so they can handle the power slowly growing without loosing control, it's probably that by collecting 99 Kishin Eggs gives a Weapon/Meister pair the experience and power they need to tackle the witch, look at Blair who was a handful to Maka and Soul but wasn't even a witch and then to Maka's expression when she saw Medusa's soul for the first time in Italy.
The final Witch Soul must give enough power to the Death Weapon to rival a Death Scythe as Death Normally chooses the target Witches not only comparing against the Weapon/Meister pair's skill but probably also the amount of power the Death Weapon will need to be considered as a Death Scythe.
Death Scythes are then assigned an area to manage in Death's place (due to his soul being anchored to Death City) probably to replace the Great Old Ones AKA Death's Eight Guardians which were once commanded by Death himself (back when he had the more monstrous look). We know the fate of 6 of the eight, Asura become a Kishin and killed 3 of them, one is in the Book of Eibon in the form of a black, amorphous blob and another is Eibon, the fate of the other 2 is unknown. Most likely before the DWMA Death's Eight Guardians managed the world so with their disappearance/fall Death created the DWMA (and became more funny looking to not scare children) and started training Death Scythes to replace them, so far there are 8:
- Spirit Albarn - North America
- Marie Mjolnir - Oceania
- Azusa Yumi - Eastern Asia
- Justin Law - Western Europe
- Tezca Tlipoca - South America
- Tsar Pushka - Eastern Europe
- Deng Dinga and Djinn Galland - Africa and Western Asia (How Wikipedia list them, not sure which is which)
It's not sure what happens to new Death Scythes with all the regions now managed but in the manga Justin Law is killed so the Western Europe slot is open and in the Manga Soul Becomes a Death Scythe.
As for what happens to the Meister that's not entirely know (at least to me), hopefully this question will give the answer to that (will edit this part when it is answered)
What about combos with 1 Weapon and >1 Meisters?
The Other way around (>1 Weapon and 1 Meisters) which is more common I only know of 2
Death the Kid: His weapons are Elizabeth and Patricia Thompson, the 2 were already guns using each other to mug people in Brooklyn and originally join Kid to exploit his wealth and authority as a Grim Reaper, but grow closer to Kid over time and outgrow their greed. Kid took an interest in them because of his crippling obsessive-compulsive disorder for symmetry however Kid wasn't a normal student.
Kilik Rung: his is Pot of Fire and Pot of Thunder, twin brother and sister, respectively, who never speak but have the ability to communicate with nature and sense danger.
I don't know of too many other Weapon/Meister pairs which have more that 1 Weapon but it's probably assumed that if a weapon has a sibling who they are very close too most likely the Meister will accept them as another weapon. I would assume these kind of weapons can be worked together but i haven't read up on too many.
As for Weapon/Meister only groups that would be very rare, in the Anime, there is a kid called Hero who didn't have a Death Weapon and was able to put up with Excalibur but ended up putting him back because of Excalibur's sniffling.
Meisters who work by themselves can manipulate their Soul Wavelength like Stein and Black Star however Stein originally was paired with Spirit, it's not known if he had a second partner (sympathy to the poor soul if there was). It's probably safe to assume that all students who do missions have a Death Weapon unless there is a very good reason for that, as it would go against one of the main reasons of the DWMA (Training Death Weapons to Death Scythes).
Death Weapons that work alone normally shapeshift part of their body into a weapon, almost every Death Weapon can do this, however Weapon duos like the Pot Twins or the Thompson Sisters can use one another; likewise with the Death Scythes.
However, pairs are normally stronger. In the Anime, Death explains how a Meister can use their Soul Wavelength alone, but it's amplified when they wield a Death Weapon (shown as Maka playing an acoustic guitar alone then plugging up an electric guitar rocking out as Soul sits on the Amplifier) so the stronger either are by themselves, the even stronger they are together − this is why Spirit and Stein are so strong together as Stein is already very powerful on his own.
As for the other away around in the Anime, Death Scythes seem to be used by others other than Lord Death; Spirit, Azusa Yumi and Marie Mjolnir mainly, however before becoming a Death Scythe Spirit was originally with Stein before partnering with Maka's mother Kami however Stein and Kami weren't his Meisters at the same time.
The other that i have heard of is Tsugumi Harudori from Not! however she befriends 2 Meisters and is indecisive about which one of them to choose as her partner which means she doesn't actually have a partner just yet. You might be able to count Excalibur having multiple Meisters, but you'd be hard press to find any who would actully admit to it.
Also there's Soul Wavelength Comparability to factor in, Black Star couldn't even wield Soul claiming he couldn't even pick him up. Tsubaki on the other hand is probably the only stated Death Weapon who could have multiple Meisters due to her accepting nature which allows for such a strong resonance with Black Star's seemingly polar opposite nature to Tsubaki, Stein mentions this when he is analyzing everyone's Soul Wavelengths during the 2nd part of the remedial class.
Soul Eater Not! just started airing, possibly shedding more light on the mechanics of that world
Soul Eater Not! follows the daily lives of Tsugumi and her friends as they learn the ropes of the DWMA and have occasional encounters with the characters of the main series so it'll probably give more light on how the DWMA/Death City works and possibly link it to how the rest of the world works, but I think it'll mostly answer question about the DWMA.