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Class C refers to the third set of numbers in an IP.
AAA.BBB.CCC.DDD

If all the links came from the same C subnet, they're 'likely' less trusted.

The 'theory' (calling it a theory because i don't think its ever really been proven) is that multiple links from the same C subnet are less valuable because its likely that they come from an owner of multiple websites.

I wouldn't label them as 'bad'

But if you're link building, it's probably a good idea to get links from different subnets

There's a lot of gray area in the widget scene.

Not all of the widget links will be considered bad - it's all about relevancy and noise.

If the site is about cars, and the widget is a car insurance comparison calculator, a link forced in the widget will likely still carry value (or at least, not bring negative value) if it's a 'car insurance' link and leading to a trusted source.

If the site is about cars and the widget is about car insurance but the link is a graphic design link, it's going to get scrapped.

I've never witnessed different results per browser.

Can you share your example keywords?

Make sure you're adding &pws=0 at the end of the url to make sure you're not getting any personalization.

Have you done any similar link building to the new url?
Is the new url 100% clean? Is the content and intent the same or very similar?

I've had similar experience where the 301 seemed to drop rank and make things worse, especially when the content doesn't match exactly. I've removed/reversed with good results, and I've also let a few incubate longer (with more link building) with good results.

If you've made an effort to build similar links, and even get some of the old links changed to point the new url and the ranking still doesn't come back - i'd remove the 301 and see what happens.