Alright, so, to start with the obvious, Caren is capable of looping in certain setups. Her skills and her weak refund mean she has relatively weird requirements, though. As a 50% battery Servant she can obviously tri-loop with MLB Kaleid, two Skadis, and any 50% charger—and she can actually do it with just a regular Kaleid and the 2004 Mystic Code if the Skadis happen to have starting NP—but her damage in this setup is Not Good. At NP2 in the 2004 setup, you’re looking at something like 60k on waves 1 and 2 (assuming you can run a backline Avenger for the extra attack buff), and 150k on wave 3. While this isn’t terrible for a plugless setup, the fact that it’s assuming NP2, it needs a backline Avenger for damage, it’s completely impossible if the Skadis don’t have accommodating CEs, it probably needs crits on wave 2 to drop a high-HP enemy, and it still doesn’t quite hit the general omnifarming threshold of 160k is somewhat telling.
Generally speaking, if you’re trying to farm with Caren, you’re going to be much better off with a CE that boosts damage and a plug buffer. If you run a 50% starting CE like Holy Night Supper or Traces of Christmas, you can get away with Reines or Waver as well thanks to Caren’s relatively flexible loop setups. For Waver, you want to have him in the front line, use all three of his skills, and then plug him out for Skadi—Caren will only get 10% in NP drains each turn, but between her batteries and Skadi’s, that’s enough (versus Knights, at least). With Reines, you want to have Caren use her own batteries on turn 1, and then plug Reines in on turn 2 (after one Skadi has used her battery) and use her AoE charge and one of her targeted charges to set Caren up for her second NP, at which point she’ll have the 10% charge on Reines and Skadi to drain and push her towards her third NP. Both these setups still have some damage issues, but they’re better than going plugless and they’re more reliable than trying to use Black Grail.
…Which, of course, brings us to Black Grail looping. Caren is unusual among Quick Servants in that she actually can loop semi-reliably with Black Grail, provided you have Castoria. With Black Grail, you can start with two Skadis, have the Skadis drop their Quick buffs and one Skadi use her defense debuff and her battery turn 1, plug Castoria in, and use Castoria’s batteries. This leaves Caren with plenty of NP to drain, and it lets her use her Quick buff and taunt on turn 2, and the other Skadi’s battery on turn 3. She needs a lot of overkill in order to do this—at NP2 and with MLB Black Grail, wave 1’s enemy HP values have to be less than 26k-ish and wave 2’s have to be 56k-ish (with some functional variance in the event of a high-HP enemy flanked by lower-HP enemies) in order for her to swing the loop, at least against Knights. These are fairly reasonable overkill requirements given recent event trends, and it gets Caren up to roughly 180k on wave 3—but it also requires plugsuit, which is relatively slow; it gets significantly less reliable with lower NP levels and a non-MLB Black Grail; and plenty of Arts Servants, along with old Quick standbys like Dantes, can hit omnifarming numbers much more easily and with much less overkill. Take that for whatever it’s worth.
That’s all assuming you’re farming, of course. For CQs, Caren is a lot more flexible, since she gets to rely on defensive NP gain and isn’t as pressed to optimize her turn counts. In CQ contexts, you can run Caren with two Skadis or in a Skadi/Castoria shell and she’ll do just fine, with her bulk and defensive utility helping to keep the team alive. Caren’s NP drains may be annoying in the Skadi/Castoria comp specifically, as it’ll make it harder for Castoria to protect the team, but aside from that she should be reasonably effective as a relatively generic AoE Quick point in CQ contexts.
More interesting, though, is her potential as a semi-support in cheaper Quick teams. While Caren is lacking in team offensive buffs (even relative to someone like Osakabehime and Jinako), she somewhat makes up for it with her defensive utility, and her plethora of batteries means you can more or less ignore her cards and focus on your DPS. You can run her in a team with one of the excellent Quick welfares—Ridetoki, Summer Yu, or Santa Karna—and either a full support (like Skadi or Castoria) or another semi-support (like Santa Night, Summer Carmilla, or Barti) for an unusually stable budget Quick team. That said, you still have the issue of Caren’s party NP drain being annoying—this is part of why I recommend welfares, as they can by default fill their NP gauges above 100% and all three of the listed Servants have batteries—and non-blitz Quick setups are one of the weakest team styles in the game, so your mileage may vary. Caren’s weak offensive buffs mean support Caren is probably more a curiosity than a practically useful Thing To Do, but if non-standard Quick teams are your jam, Caren may have some value to you.
I more or less covered CE options for Caren in the looping breakdown above, but to recap: your options are Kaleid for simple-but-low-damage looping, 50% charge CEs for a mix of consistency and damage, and Black Grail for maximum damage if you have Castoria and can swing the overkill. Outside of farming, Traces of Christmas is a great CE pick for both general-purpose DPS use and for support use, as it lets Caren NP immediately with just her own skills on top of boosting her NP gain some. Black Grail is also great for damage purposes in CQ contexts, especially since it functionally boosts Caren’s attack by 20% as well thanks to her weird unique passive. If you’re leaning hard into Caren as a support, Prisma Cosmos and 2030 are good picks as well—2030 for general crit consistency, and Prisma so you can pretty much just ignore Caren’s cards and focus on your DPS.
Command Codes that boost Caren’s crit damage will be nice picks in general, as Caren’s lack of innate crit buffs (aside from Independent Action) means they’ll have a more noticeable impact than they do on a lot of other Servants. Healing CCs are also especially nice on Rulers, as the healing is effectively doubled unless you’re fighting a Berserker or some such.