I've been told by native speakers that the former refers to little dogs while the latter is for bigger dogs. I also have been told that the latter has a negative connotation. I'm wondering:
There's no negative connotation, at all.
条 and 只 are used interchangeably in today's Chinese, not just specific to a region. Long ago, 头（頭）were used as a quantifier for dogs or other farm animals. I suspect "head" was dropped because dogs do not have stocky builds as other animals(pigs, donkeys, bulls etc). Also "头" tends to associate "dumb animals" and dogs are perceived as intelligent.
条 in most cases is used to quantify something that's slender, e.g. 一条路(road)， 一条河(river)，一条虫(worm).
I say 一只狗 because it seems to make more sense considering others animals of simliar size and stature also use 只. e.g. 一只猫，一只狐狸 etc. But this is just a personal preference.
I'm afraid I cannot agree with you (or with the native speakers who told you so).
The expressions don't have the differences mentioned in your posts. To me they are equivalent. I prefer "条" though, for no reason.
In most of context, those two just exchangeable. 只 and 条 is measure word, sometimes 只 means 'little' and 条 means 'long' and maybe 'big'. for some person,esp. girls, 'little' hint 'lovable'.
Probably you can find the negative meaning of 一条狗 from 周星驰 (Stephen Chow)'s movie 《大话西游之仙履奇缘》：
“那个人样子好怪，他好像一条狗。”
That person looks strange, he looks like a dog.
or
“你看，那个人好像一条狗哎。”
You see that person? He looks like a dog.
"一条狗" here indicates they thought that person is a "poor guy" rather than "a dog".
But don't use this if you're not familiar with the people you're talking to!
In most cases, we just make fun of our friend or our self that we use "一条狗" to refer to some difficult situation, and have no negative connotation at all, for example:
在上海工作，一个月收入两万元，活的还不如一条狗。
I'm working at Shanghai, and I get paid 20,000 yuan each month, and live here is not better than a dog's life.
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