I can believe, in theory, that the location extension point is possible - even if the organic algo doesn't take your direct paid search data into account, they may share data sources, including location data (since that shared data doesn't bias one system based on the other). That said, I've met way too many AdWords reps that had no idea what they were talking about and yet, somehow, were happy to spout off misinformation. No offense to your particular rep, but again, I've had too many bad experiences. It's important, for many reasons, that your location data be consistent and accurate, but I don't think location extensions are a huge cause for concern.
There is an argument to be made, though, that working with the rep has benefits, even if the rep isn't always right. Reps tend to dig in, and if you aren't solving something they think is an issue, they may not help you get to other issues or positive traction. I recognize, though, that it can be hard to work across multiple parties, and your options aren't always ideal.
The off-domain landing page issue is a lot more complex. One common impact we see of paid search is the indirect boost it has on organic, based mostly on perception. If I see your brand in organic and the same brand in paid, and if the message is consistent (and, of course, appeals to me), my odds of clicking one or the other go up. This isn't an algorithmic bias, but a human one. Broadly, it also means that your paid ads are contributing to brand-building and name recognition.
If you're using a separate domain and if that domain is visible (and especially if it's confusing) to users, then you may be cannibalizing yourself. It's especially problematic if you've got two domains sharing a common message - this could confuse searchers about which domain is the "real" representative of your brand.
As Andy said, it also means that any links/mentions/etc. you might get to that secondary domain aren't helping you. In most cases, it's rare for ad landing pages to accrue links, but if a landing page is integrated into a site, then people are more likely to click through to other content, which might be shareable, linkable, etc.
Unfortunately, all of this is very speculative, without knowing more about your particular case. I think it's just important to note that there are many indirect interactions between paid and organic.