If your character is busy doing something as a 1 round+ action, are they able to still take advantage of AoOs?
No, you can't. Because you don't threaten enemies
Core Rulebook, Combat, Full-Round Actions, Cast a Spell (page 187) says:
You only provoke attacks of opportunity when you begin casting a spell, even though you might continue casting for at least 1 full round. While casting a spell, you don't threaten any squares around you.
Since you are casting a spell for the full duration of the round, you are not finished casting the spell until the start of your next turn (possibly longer depending on the casting time). This means that while casting a spell that takes 10 minutes (ie: Divination), you cannot make Attacks of Opportunities for 10 minutes.
Attacks of opportunities are usually provoked when an enemy does certain actions while inside your threatened area:
You threaten all squares into which you can make a melee attack, even when it is not your turn. Generally, that means everything in all squares adjacent to your space (including diagonally). An enemy that takes certain actions while in a threatened square provokes an attack of opportunity from you. If you’re unarmed, you don’t normally threaten any squares and thus can’t make attacks of opportunity.
I would suggest the answer to be NO, because you are expected to be concentrating on your spell not looking around for someone to attack as it moves past you. You cannot really focus on the intricacies of A spell if you are moving about and attacking. Especially if that spell has a somatic component.
Simply put, you are asking to take a separate action while you are already occupied by an action that requires all of your attention.
Yes (but beware of 1-minute casting time spells, and you need special threatening abilities)
Casting a spell makes you no longer threated spaces 'around you', so you'll need some other method of threatening spaces or some other method of making attacks of opportunity. You can still threaten spaces with anything that adds threatened spaces in a special way, like the snap shot or whip mastery feats, and possibly with reach weapons depending on how you define 'around you'. You can also make attacks of opportunity without threatening spaces with the Vicious Stomp or Misdirection Attack feats.
Regardless, your weapon will also need to not be two-handed or you'll need extra arms if your spell has somatic or material components.
What's much more complicated, however, is the question of whether or not choosing to take an attack of opportunity while casting a spell, when you can, prevents you from casting your spell.
The relevant rules are as follows:
Most spells have a casting time of 1 standard action. Others take 1 round or more, while a few require only a swift action.
A spell that takes 1 round to cast is a full-round action. It comes into effect just before the beginning of your turn in the round after you began casting the spell. You then act normally after the spell is completed.
A spell that takes 1 minute to cast comes into effect just before your turn 1 minute later (and for each of those 10 rounds, you are casting a spell as a full-round action, just as noted above for 1-round casting times). These actions must be consecutive and uninterrupted, or the spell automatically fails.
When you begin a spell that takes 1 round or longer to cast, you must continue the concentration from the current round to just before your turn in the next round (at least). If you lose concentration before the casting is complete, you lose the spell.
So, these rules are weird. First of all, the specific calling out of spells that take 1 minute to cast is very odd. Technically this only affects those spells and makes casting them much more onerous than casting a spell with a 9 round or 11 round casting time (not that any exist. The closest spells are Forbiddance with a 6 round casting time and potentially 2 minutes (or maybe fractional minutes) for Illusory Script, Nature's Ravages, and Ironwood on the other end), since all other spells require only a full-round action to start and then a standard action each round.
There's a couple ways this is ordinarily dealt with. One is to extend the rules for 1-minute spells to all spells longer than a round, and the other is to completely ignore the rule and have 1 minute spells cast the same way as everybody else.
Regardless, there's a second complication here. Originally, in the Core Rulebook, attacks of opportunity were never given an action type. Maybe they were free actions, maybe not-an-action actions, maybe their own special type of action. It was never defined, so it was sort of up to the GM how they wanted to handle that.
In the Melee Tactics Toolbox, however, Paizo errataed/clarified attacks of opportunity to definitely be free actions. Paizo's errata rules are last published wins, unlike WotC's primary source rule, so even if you interpret the (re)definition as conflicting with the CRB rather than expanding on it, technically the more-recently-published Toolbox wins (2015 for the Toolbox, 2013 for the most recent 6th printing of the CRB). That makes AoOs a free action, but that's hugely problematic for certain interpretations of the free action rules, so this also gets ignored very frequently. This means there are four common cases which need addressing, based on the rules as written for the two sections, the most common non-RAW method of play, and their combination.
Regardless of rules interpretation, 1-round casting time spells are always okay to interupt. This is because they only need a single full round action so there's no concern about making that action non-consecutive with other actions. Similarly, it is always okay to take an attack of opportunity in the final round before you finish casting the spell, because you'll have taken all your full-round actions by that point.
It's also okay to take an attack of opportunity in response to an attack of opportunity that provokes which is itself taken in response to your starting to cast a spell (e.g. you start casting Forbiddance, someone with a -2 cursed sword tries to punch you in the face, you punch them in the face first). That's because AoOs happen right before the action that provokes them, which means that opportunity attack you take goes back in time so the full-round action you took before it is now after it and everything can still be consecutive.
It's not okay to take an attack of opportunity in the middle of casting a spell, because even as a free action it makes your full-round actions non-consecutive. You still can, if you want, but then your spell will automatically fail. This is true unless attacks of opportunity are not-an-action actions, which, since they aren't actions, can't interrupt your full-round action sequence and make them non-consecutive or the rules for casting 1-minute spells are not extended to all other spells longer than one round and the spell you are casting does not have a casting time of exactly one minute, or the casting time rules for one minute casting time spells are ignored.
In those cases, it's always okay to take an attack of opportunity.