1) Link to them from several different categories? (e.g., Best Selling Widgets, Silk Widgets, Japanese Widgets, Widgets Under $50)?
Absolutely. Think of your upper level categories as being equal to your home page.
2) Link to them directly from a global footer or global header?
No.
3) Link to them using in content text from only closely related products?
Yes. For example, "Related Products" - this helps to spread the love usually in a horizontal fashion if products are cross linked within their own category.
4) Slap noindex tags on pages for other products / categories that you sell, but don't rank well at all, and that you won't have the time or resources to try to get to rank well?
If you don't have the site equity to support those categories, you could do that. But, most of the time that is not the case unless it is a new site.
5) Something else?
All new products need to go through a process. They should first appear in a top level category, possibly a What's New section for that category. They should remain there for a period of time after which there should have been enough crawling, juicing, to "seat" that document. Repeat and rinse.
Treating your categories as stand alone sites is probably one of the better ways to think about this. Internal linking plays a very important role in all of this. You need to be frugal with the number of links in any given area. Specific areas are where the majority of the juice flows, those need to be focused on.
Links that are in headers and footers may not have the weight they should have due to their repetition. But, if you have headers and footers that are dynamic and specific to each section, then you may be able to finagle something. ;)
Personally? I like to segment categories as much as I can. I try my best to make sure there isn't too much disconnect with the navigation menus. We typically serve menus dynamically based on the category. The header menu may change, the left menu definitely changes (if applicable) and the footer may change.
The goal is to make those categories independent and not too reliant on the root. They should stand on their own. The root is the front door (Storefront). You want your best at the front, and then you want to make sure once inside, that visitors are directed to your best (first) in all parts of the store (Aisle Presentation). Products that sell the most, get the most shelf visibility. Products that don't, get spacing on a shelf towards the back of the store. Every now and then, those products end up on Sale (Clearance or Sale Category). They remain there for a little while, enough to get juiced, then you put them back. Rinse and repeat.
This process applies whether you have 100 products or 1,000,000+. That "horizontal plane" becomes much wider as the number of products increases. The pyramid becomes fatter. Within the big pyramid, are smaller pyramids. Some are even upside down with the document at the tip of the pyramid (at bottom) being juiced heavily. Follow me?
Let's not forget about host names either. ;)
