Are there particular districts that you know of that are both easier to get into than others and would be looked upon favorably by a NY V10 firm
Virtually any Article III clerkship would be looked on favorably by virtually any V10 firm, IF you want to be a litigator. I'd suspect that the views of a USDC clerkship would vary firm by firm (and maybe group by group) if you want to be a transactional attorney. Some would look at it as a frivolous waste of time; some would see it a prestigious resume booster; and some wouldn't care. But that's all pure speculation.
No district court clerkship is "easy" to get these days. Even the proverbial DSD* clerkship is probably going to get 300+ applications. But some are definitely harder -- SDNY, CD/ND Cal, DDC, D Mass, ND Ill, maybe one or two others.
Is applying for a clerkship the kind of thing where, e.g., I would be really happy to clerk at the USDC in South Dakota, but because judges have never hired students from my school there, my application would not be reviewed so it wouldn't be worth applying
Some judges are like this, some aren't. My general intuition is that as the clerkship market becomes super-competitive, you get a lot fewer hire-only-the-local-law-school-grads judges these days -- even the judges in (say) Wyoming realize that they can hire top 1/3 students from HSY, top 20% students from CCN, and top 10% students from the rest of the T14 (plus the top grad from Wyoming). So even if they used to hire only locally, they don't now.
* By the way, if the DSD clerkship was in Rapid City, it would be awesome. The Black Hills area is really pretty, and the weather is a bit better than the rest of the state.