My information may be inaccurate, but doesn't Shirou technically have 20% of the Emiya crest lying around somewhere? At least, I understand Kiritsugi had access to it. And it wouldn't be too far a notion to think that after the war, Shirou discovers some old manuscript or diary detailing how to get whatever of his legacy is left.
Even if the Crest was untainted after being splashed by Angra Mainyu, even if Kiritsugu hadn't destroyed it, even if it hadn't decayed from being underground attached to a rotting corpse for 10 years....
The Emiya Crest has no relation to Shirou. It would be like transplanting an unrelated stranger's kidney into your back. It might work, but 99% likely it would be rejected. That applies to Shirou, and his children.
As for the name, the kid would be the Tohsaka heir, so the name Tohsaka would be more likely, all things being equal... but traditionally a kid does take their father's name. Considering Japanese culture, though, it wouldn't be unreasonable for Shirou to be "adopted" into the Tohsaka house as a son.
In the end, I think it's more likely the child of Rin and Shirou would be "Tohsaka". I think it would be better storytelling to have a kid named "Tohsaka" that has complicated feelings about the history of that name, rather than simply shoehorning the name "Emiya" in because that's what you 'want' to do.
It's not theoretically impossible that the kid would have the same Origin as Shirou, but for the purpose of your story, yes, it's impossible.
Shirou's origin is already wonky because of his screwy past; most origins are actions, not things, for one thing, and for another it's highly individual. It's like the purest core of your personality. And figuring out what your Origin is... isn't easy, or even necessarily wise.
Elements are which of the five elements--earth air water fire ether--you can use in your magecraft. Exceptional elements exist; Sakura's "Imaginary Numbers" allows her to interact with phantasmal existences pretty much directly.
So if elements say "what kind" of magecraft you can use because of inborn ability, origin is "how and why" you use it because of your inclinations.
Transforming gems into swords... just sounds lame. It's too next-gen copypasta.
If I wanted to express what a magus raised by Shirou and Rin would be capable of, I would start with a statement like "good with flowing mana into gems (Tohsaka) and good with physical magecraft (Shirou)". Well gems are mineral crystals, with a covalent bonding pattern; and metals are metallic crystals. So something like "using Tohsaka gemcraft on steel" would be obvious. Because it's magic, I'd say that metal can probably only store a tiny fraction of the mana a gem could. But, if that tiny fraction of mana is directly injected into the enemy's body because the metal it was soaked into was a bullet, that could still be an effective weapon.
I guess I'm saying take a wikiwalk and come up with something non-obvious but with a clear lineage in retrospect. If you absolutely totally have to give this kid a special ability, make it special, instead of just combining two monsters in Monster Rancher to get a blended monster.
In the end...
I don't want to read a story about MOAR BATTLEZ when everyone involved is an OC. It could work, but... not the way you are presenting it.
This story should be about the relationship between these family members. If the child is getting into BATTLEZ because they're chasing after dad, and mom and dad have complicated feelings about that, that could be good.
When you posted the OP, I got the impression you wanted to write something like "Rin drags Shirou away from his hero-complex with a child" which would be a little more "soap opera" than I usually advise, but it could work.