M.B.
Given that the problem in your kitchen is sporadic, it seems to me that you have a loose connection somewhere OR rodents (squirrels, mice, etc.) may have chewed wires in your attic. They are well known for doing so. You might want to check and see if any of your phone line is run in the attic. The fact that it seems impacted by the weather seem to imply that you have an undue amount of wind entering your house, as well (if it is inside).
How did Verizon determine the problem is inside the house? Did they test the line that comes to the house? If not, whomever you spoke to may have just made that assumption based on your description that some jacks work and some don't. It is still possible that you have a phone line that is inground that might be knicked or if it connects aboveground, that might also explain the weather impacts.
I haven't had the need to call the phone company for these sorts of problems for many years, but my understanding was that if they come to check it out and find the problem is in the house, then you pay the service fee (if you're not paying that inside maintenance that another poster mentioned), even if they don't come inside to fix it. But, if they find the problem is outside, then it is their problem. Of course, this may have changed in recent years.
I had a similar problem with my cable as where it split in the attack, the connection was not done correctly and my Internet access would go out in the afternoon as the attic heated up. With the cable company, the situation was similar in that if it was a problem with my equipment, then I had to pay the service fee. But, since it was with the basic connection in the house, there was no charge.
Another, perhaps cheaper overall option is to switch to Internet phone connectivity using something like Vonage.
