What is the year this universe is taking place?

In the currently being written story the date is 2045. Later we might open the time line for others.

Is this an open universe?

As of right now it is not. BUT, we might after the first stories are written.

When was the first Bio-Morph made?

The first Bio-Morph was an accidental conversion of the young woman who was developing the virus in 2019.
She accidentally exposed herself without knowing it until after she also exposed herself with activation enzyme.
Unlike later conversions, hers was slow and painful. She was also a few weeks pregnant at the time so her son ended up being the first natural born Bio-Morph.
Due to a clause in her employment contract stating, 'all her research and work is owned by the company', she and her son became property of her former employer.

What is the name of the company that she worked for?

Yakeru Bio-Tech.

How did their slavery start on the national level?

In 2021 to 2022 a major electronics manufacturer in china converted their low wage factories to robotic ones. This manufacturer supplies the bulk of china's exports.
At the same time in Japan the government bought special robots and outfitted a retirement high-rise with them. Due to demographic shifts fewer people were taking care of the elderly.
In less than a year the plants in china have failed to produce the volume needed to keep exports up and it is hurting china's trade balance.

A few months later due to an unforeseen bug in the code of the elderly care robots was discovered.
One of the robots after being loaded down with orders by the old man it was charged with tried the best it could to execute all of them.
One of the orders was to cook him a meal, the robot started it, and then it wasn't able to return to the kitchen to finish it as the old man loaded it with more orders.
The food sparked a fire that spread throughout the building. To make matters worse some of the robots wither hindered the resident's escape attempts or hindered the rescuers.
It did not matter to the Japanese public that the bugs were simple and fixable, the view that they were unsafe was cementing in their psyche.

Both incidents caused black eyes for their respective countries. Yet at the same time Yakeru Bio-Tech was refining the conversion virus and Bio-Morphs in general.
They saw an opportunity in the Japanese tragedy and the Chinese failure, and lobbied the respective governments to use their new Bio-Morphs instead.

Okay, but how did the United States and the rest?

A major fast food chain at the same time was struggling with their under performing automation systems that they bought to replace workers. Workers who dared to want a living wage.
Once their stock skyrocketed from the return to their former performance level, just without paying workers. The rest of the nation's companies soon followed steamrolling any attempts to ban the practice.
Other nations followed suit more or less so they can compete.

You mentioned Mexico and Canada have given human rights to Bio-Morphs. Do Bio-Morphs try to flee to those countries?

Yes if they know about it.
There are also underground rail-roads for Bio-Morphs going to Canada. Mainly because Canada was not forced to sign a Bio-Morph return law.

** How many Bio-Morphs are there?**

Exact numbers are hard to come by, but estimates for the U.S. put the population close to one hundred million.
This is though only for native born and converted, estimates from china are non existent and other countries have stopped counting.

How intelligent are Bio-Morphs?

This depends entirely on where you live.
In the United States and pro-slavery Europe all the information you will find on the locked down Internet will state that Bio-Morphs, while more intelligent than animals, are dumber than the lowest I.Q. human.

In china simply asking publicly flags you in their intelligence system.

In the rest of the world you will find that they are well within the human spectrum of intelligence.
The highest I.Q. Bio-Morph is about ten percent lower than the highest I.Q. human.

Yet they have a few burdens that humans do not.
Instincts are much stronger in them than in humans. For example a female Bio-Morph can be a P.H.D. researcher, yet when heat comes around in mid fall for a single week all she will care about is mating unless she takes the appropriate medication to mitigate it.

You mentioned a locked down Internet, whats with that?

In the United States and other non Asian nations they used the organized Bio-Morph rights movement to lock it down to prevent further 'terrorist' actions.

The U.S. named it 'The Authorized Network Communications Act of 2040' or TANC(pronounced tank).
The result was that coder's and websites have to be registered with the act and everyone has to have a national id on-line.
This id is used to track where you go, what you post, what you buy, and what you watch on-line.

Client computers have to have the newest 'trusted computing' system and 'approved' O.S., read windows, installed to even connect to an I.s.p.
To browse the limited web, a user will also have to use a browser that is certified by the act.

As a side effect the old B.B.S.(bulletin board system) that existed before the Internet has experienced a renaissance.

Is this a mundane, semi-mundane, or fantastical universe?

Pure mundane at the current time.

So how does the United States and other western nations deal with the large unemployable population?

A nominal tax, roughly about 1%, on all Bio-Morph and Bio-Morph related transactions.
The money in the U.S. is used to purchase and run jobless dorms made from; Old hotels, motels, unused schools, malls, and in some cases just plain land for tent cities.
These places house, feed, and cloth people who are unemployable, and their kids who are unemployable by proxy.
Because no one hires anyone who lives here.

In the UK it is used to keep people in their homes, and in France it is used to provide re-training to work in the Bio-Morph industry if possible.