Not sure why no one has addressed this. My own qualms are that I am not familiar with codes wherever you are, and it is up to the AHJ in any case.
However;
1. Probably, depending on where you cut into the stack and what sort of floor you have. The throne may need elevating to satisfy slope. When I say "may" that means I can't tell from here.
2. Yes, if it is above the roof it penetrates. But I'm wondering if you had to put some turns in your stack to get there, which starts "if" questions.

Talk to your local Code Office.

Under the 2000-2006 International Residential Code plumbing provisions, there is no limit on how far a water closet (toilet) can be from its vent.

BUT...

However, The distance a sink trap can be from its vent is limited by the size of the pipe that serves the sink trap.

For instance, if the trap is 1 1/4" diameter, the maximum distance the vent can be from the sink trap is 5 feet...

If the trap is 1 1/2" in diameter, the maximum distance the vent can be from the sink trap is 6 feet...

If the trap is 2" in diameter, the maximum distance the vent can be from the sink trap is 8 feet....

Other plumbing codes have similar restrictive provisions.

What you have proposed for you sink would be illegal and a plumbing code violation almost anywhere...

I can't believe that a couple simple questions that should be nothing more than a silly pop quiz for a first year apprentice is so difficult to answer.

The post states that he is adding a 1/2bath in a basement and the proposed fixture location is 15' from the stack.

Under the IRC the maximum allowable length of an un-vented fixture arm can be mathimatically computed by dividing the diameter of the line by the required pitch. By example bathroom lavatory requires an 1-1/4" line with a 1/4" per foot pitch so the maximum length would be 1.25 / .25= 5'. The code will permit increasing the size of the line by one nominal trade size, which means they could make the lavatory drain line 1-1/2" and that would allow 1.5 / .25= 6'.

Code requires a water closet to be on a 3" line, and using the formula for a 3" line it could be 3" / .25= 12' however, there is an exception in the IRC which states that if a water closet is the only fixture served by the 3" line it may run an indefinite length.

Under the UPC the length limits for un-vented fixture arms are considerably less. I.E. 1-1/4"= 2'6", 1-1/2" = 3'6", 2" = 5', 3"=6' and 4" or larger =10'.

However, even if the fixture locations were within the prescribed length limits for an un-vented fixture arm it would still be a code violation because the existing stack is obviously serving a water closet on the upper floor, and you may not connect an un-vented fixture arm below a water closet.

On the other hand, there is a very simple solution.

As alphonse previously pointed out, the first step is to determine the depth of the existing main drain line where you intend to tie in.

At the new water closet location you will have to install a closet flange and a 1/4bend to transition from vertical to horizontal. Consulting the Charlotte Pipe Co DWV fitting catalog we find that the required fitting allowance for a 3" 1/4bend would be 8". The required drop for a 15' 3" pipe would be .25x15= 3.75 or 3-3/4" thus the bottom of the existing main drain pipe at the point of tie in must be 8" + 3-3/4" = 11-3/4" below the finished floor level.

Assuming that we have enough depth at the point of tie in, the remainder of the problem is totally elemental.

Please note that in the original question Jockey16 was asking about the code requirements for an auxiliary vent. Once an auxiliary vent is installed within the aforementioned length limits all the remaining pipe downstream from the vent is classified as a vented branch and may run an indefinite length and even though code prohibits attaching an un-vented fixture arm to a stack below a water closet there is no restriction on attaching a vented branch.

The question then becomes, what is required to create the auxiliary vent?

under the IRC an auxiliary vent may be reduced to 1/2 the diameter of the line it serves while under the UPC all vents must be sized by the DFU (drainage fixture unit) load.

The water closet requires a 3" line so under the IRC the vent must be a 1-1/2".

Under the UPC the DFU load is 3dfu's for the water closet and 1dfu for the lavatory for a total combined load of 4dfu's. Under the UPC a 1-1/4" line is limited to one dfu but a 1-1/2" auxiliary vent is rated for up to 8dfu's so here again the proper size for the auxiliary vent under either code is 1-1/2".

Code requires a cleanout on the upstream end of all DWV lines therefore you begin by running a 3" line straight from the water closet to the stack. (A water closet is removable allowing direct access to the end of the line, therefore it meets the requirement for a cleanout.

Next install a 1-1/2" riser directly behind the lavatory and run the riser up as the new auxiliary vent. Install a Tee on the riser to extend a fixture arm out to the lavatory trap. At the base of that riser install a horizontal line over to the 3" from the water closet and tie into the 3' with a Wye & 1/8th bend or combo (A tee is prohibited here). If you are under teh UPC te 1-1/2" must connect to the 3" within 6ft of the water closet. If your under the IRC there is no limitation.

Code prohibits running vent horizontal until the vent reaches an elevation at least 6" higher than the flood level rim of the fixture it serves, however in this configuration the line extending up from the tee behind the lavatory is defined as a vent while the line extending down from that tee and horizontally under the floor is technically defined as a "Combined Waste & Vent" and it is legal.( The reasoning here is that if we have a dry vent running horizontal under the floor there is a strong probability that solid particulates would back flow into the pipe and it would become obstructed in a short time whereas in the combined waste & vent configuration when liquid is discharged from the lavatory it will wash the vent line.)

At this point we have figured out the drain requirements for both the IRC & UPC so let us now turn the the second question, how to terminate the auxiliary vent through the garage roof.

Once the vent line rises at least 6" higher than the flood level rim of the lavatory bowl it may be offset horizontally however under the UPC if the horizontal offset is greater than 3' the entire length of the vent must be increased by one nominal trade size, which means we would need to install a 2" line rather than the 1-1/2" as specified.

Under the IRC you must increase one nominal trade size if the total developed length of the vent from the 3' drain line to the roof termination exceeds 40ft.

If you live in a region subject to frost the the vent line must be increased to 3" diameter at least one foot inside the structure before penetrating the roof. (Check your local code, some codes require 4").

The tip of the vent must terminate a minimum of 6" above the roof line when measured from the high side of the roof pitch. In regions subject to snow it must be 6" higher than your average snowfall (Check your local code office).

The vent must be placed a minimum of 12" from any vertical surface. I.E. 12" away from the gable wall on the upper floor or a chimney.

Under the IRC the vent must be 2' below any structural opening that is within 10' horizontal. It must be 4' below any window within 10' horizontal.

All roof vent penetrations must be fitted with an approved pipe flashing.

All vents must terminate a minimum of 7' above any roof used as a deck.

Thorough and very helpful answer!