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B. Can having my link program in a subdomain of the main domain cause any problems?
Everyhting about your site should be first and foremost provided for the benefit of your readers. If it benefits them, it's good. If it doesn't it's bad. After that, and within that you can make tweaks to encourage the search engine spiders to recognise the nature of the page and rate it accodingly.
The base href element simply lets the search engines know the absolute URL for resolving any relative links. It does no harm whatsoever, and is generally good practice when relative URL's are used.
As for Point B, as soon as you say "link program" I wince, since reciprocal links are increasingly discounted by the SE's. I suppose having it on a subdomain might throw off the SE's in case they were inclined to discount all these links, but since a subdomain is still clearly affiliated to the base URL, I doubt there's much benefit in doing things this way. In fact, it looks sneaky to an extent, and might be an additional mark against you in the event of a hand check.
Whoever you link to, just do it from your own site, and try to avoid excessive reciprocal linking. More and more, you gotta get quality one way inbound links to rank well.
My 2 cents anyway.