Yes it does. And here's what I've understood about sDNA. Spiritual DNA is kept in the Spiritual Realm, stores certain information about you. It says what planet you're from, what your level of connection to various Shards are, and can be passed down from generation to generation. What Lerasium does is it modifies your sDNA so that it gives you a +10 connection to Preservation, thus creating the side effect of you becoming an allomancer. Lerasium slowly is diluted as that effect is spread from generation to generation, just like one person with pink hair will only have so many descendants with pink hair.
Scadrial is the only world that we know of where magic systems (Allomancy & Feruchemy) are passed through sDNA.
As on the idea of Focus on Roshar, I think that Odium's Shard's Post up above is the right answer. The focus in all the other systems have been something that has determined what form the "magic" displays. Aons, Commands, and Metals all have a function. I do believe, however that there is some give in each focus. For example, Iron doesn't make you pull on all the metals around you. It instead lets you choose which metal to pull on, with how much strength. Brass gives you even more variety. You can choose someone, and soothe specific emotions.
The same of Spren. Syl is Kaladin's focus. She allows him to access the Stormlight, through the gems, and use it to manipulate bonds. He can manipulate the spiritual gravitational bond like Szeth does to go flying, and make things go flying, or he can manipulate the electromagnetic forces to make things "stick".
Now on to Gems. I believe that Gems on Roshar are like plants in our world. They take a raw form of energy from the Highstorm, (like the Sun), and converts it into useable energy, Stormlight (like glucose). The trouble is, only certain organisms can process Stormlight (glucose)- Spren (Mitochondria). The Spren can then fuel that power to those that they share a Nahel Bond with. (Just like a mitochondria gives energy to the cell that's hosting it.)
Or, if they are instead bonded to the crystal, they perform a certain defined function, using the Stormlight as their fuel. I know that for Soulcasting, specific gems mean specific transformations, but if you change the spren inside of a gem, you presumably change the Fabrial as well. That explains the variety of things that Fabrials have been found to do.
(Note that I speak of mitochondria with the theory that they were once independent bacteria that were eventually incorporated into successful cells. This may be wrong, but it's just the idea that matters for this theory).