I feel bad for not chiming in earlier - school started on the 11th, and I missed this entire conversation because I only had a few minutes here & there to check on this forum during that time.
I am willing to bet you got selected for verification AND you had a 399 code on your FAFSA. This code, used during this year of using 2015 income a second time, results when income info reported for 2016-2017 differs from income info for 2017-2018 (it should be the same, since both years use 2015). The aid office is required to verify income for BOTH years, and they must make changes for both years. Since it sounds like your income documentation issues were not resolved before the verification deadline for 16-17 passed, there was nothing that could be done to change the 16-17 Pell issue.
The real bottom line is that the parents must complete the FAFSA correctly, and the student must complete the FAFSA correctly. To blame an aid office if the FAFSA was not correctly completed is pushing the blame where it doesn't belong. If the FAFSA is selected for verification, it is important to collect the required documents quickly - and yes, the issues with the IRS this year did really mess things up for both students and aid offices where this is concerned. The aid office was required by law to collect the documents that did not come quickly, so it's not that they wanted to mess with you. Sometimes, unfortunately, the aid office will receive documents that raise more questions, and they then must collect more documents ... they are required by law to resolve any conflicting information. It's not that I don't have sympathy for your situation, but I do know what it's like to deal with a messy situation like yours ... rules cannot be bent, even if the aid officer wants to be helpful.
Yes, the year round Pell is included in the lifetime limits. The caveat that the 101-150% must be at least half time is intended to help people to avoid running out too early. If students are taking at least 6 credits for that portion, the idea is that they are hastening graduation (not just taking a class here or there).
If you enroll in 12 or more credits in a semester, you receive 50% of your annual award. Say your Pell is $5,000 ... you would get $2,500 per semester if you enroll in 12 or more credits. It's going to be the same in summer. If you enroll in 12 or more credits, you receive 50% of the annual award, or $2,500 in the example. If you enroll in 9-11 credits, you receive less ... 37.5% of the annual award. For 6-8 credits, you get 25% of the annual award. BUT ... you won't get Pell for less than 6 credits in summer ... UNLESS you didn't use your full 100% annual award in fall/spring (that is, you were enrolled in less than 12 credits in at least one of those semesters). In that case, you can receive any remaining Pell from the first 100%, up to 12.5% of the annual award amount.
You don't apply - your school is required to award it to you if you qualify. Here is the info from the Department of Education website: To be eligible for the additional Pell Grant funds, the student must be otherwise eligible to receive Pell Grant funds for the payment period and must be enrolled at least half-time. For a student who is eligible for the additional Pell Grant funds, the institution must pay the student all of the student's eligible Pell Grant funds, up to 150 percent of the student's Pell Grant Scheduled Award for the award year.