Three Years and a Day?

I'm doing some reading for class in Lee I. Levine's Sages, Patriarch and Community Life, which has provided some interesting fodder on parnasa (i.e., charity livelihood/making a living/money required to live), which happens to be the topic of my semester-long paper framed around Ovadiah ha'Ger (after all, the man survived in multiple countries on the charity of Jewish communities, and charity/tzedakah/parnasa is sort of a major theme of Jewish living). But what I want to throw out to the crowd here is something interesting that is sort of mentioned in passing but not really discussed. Of course, being me, I sort of stopped, read it over again, read the paragraph after the quote, and wondered "what?! why!? huh!? is this actually halakah?!" So here, let's start with the quote.

R. Jacob b. Idi (said) in the name of R. Joshua b. Levi: "It happened that there was a family in Rhodes that was being questioned (about its priestly lineage). And Rabbi sent R. Romanus to investigate. He checked and found that an ancestress had converted (to Judaism) before she was three years and one day old (and had subsequently married a priest). He then declared it (i.e., the family) eligible for priestly status (BT Yevamot 60b).

So here's what resulted in my pause: the law states that a convert cannot marry a kohen (i.e., a priest), and her daughter may not marry a kohen, but that the convert's granddaughter and further generations may marry a kohen without question. In this instance, a woman converts at a very young age, marries a priest/kohen, and the family many generations (we assume, no clue what "ancestress" means as far as years gone by) later is declared kohanic.

My questions are

1. What does three years and one day old have to do with anything? I've never seen this limit before.
2. Because I've never heard of this rule, I'm perplexed that she would have been allowed to marry a kohen. 
3. Because I've never heard the rule and because I'm perplexed that she would have been allowed to marry a kohen, I'm confused as to why the family would be declared kohanic. 

So, if you're a master of the laws on conversion and can explain No. 1 and, as a result, Nos. 2 and 3, maybe I'll feel better about this little anecdote. 

Sidenote: "family" is found elsewhere as "maidservant" and elsewhere again as a "town." Levine says it's unlikely that a special envoy would be made for a simple maidservant (also, what would it matter if she was a bat kohen?).

Another note: Okay, I used the wrong translation for parnasa but you have to understand in my head it's all one big ball of sense-making. If that makes sense. For some people, parnasa is how they survive -- like Ovadiah haGer. For him charity/tzedakah WAS parnasa!)