Wooooooooooowwwwwww! So the summary made this sound awesome the concept is cool and certain scenes are fun, sweet, funny, or interesting but it does have several problems. It's been years since you updated but you might continue this one so here's what I noticed.
Grammar: your spelling seems fine but your sentence structures tend to be wonky. Occassionally you repeat a phrase or action someone is doing close together in a weird way. Example would be earlier in the story there was a part where Naruto was looking around and spoke and then you described him looking around as if he hadn't already been doing that. There is also details describing actions that don't flow well.
Continuity: you seemed to do well until the further you got in the story. Examples being Lucy being wary/slightly afraid of Mariko due to "rumors" and how she saw her treat the people giving them hell(and why should she care about that when she wanted those people dead?) yet having no issue with engaging in a pillow fight with the child and giving her almost motherly affection despite this supposed fear. Next is Kurama's annoying ability to not die despite all factors pointing towards how he should've died from the explosions or the attacks aimed at him or from the fact he's making himself a nuisance as well as a threat and if he's not dead he should've been captured because he's not faster than a frickin magical ninja. After that is the unexplained mystery of how that organization can just continuously infiltrate legit ninja villages when they shouldn't be able to, move around unnoticed when they shouldn't be able to cuz they stick out, quickly find the girls when they shouldn't be able to as they didn't yet have tracking chips or they were destroyed so exact current location can't be known, and leave with no obstacles when AGAIN they shouldn't be able to. Especially when the villages are aware of the last security breach/about them and would've upped security. Security that should already be high cuz they had other threats to be concerned with. Next is Sai. Perhaps I misunderstood but you'd already established Sai as a teammate both Sakura and Naruto knew from both a previous mission where they had to intercept Sasori and Deidara as well as one where they dealt with bandits and Sakura was trying to argue with Sai over what to do. But then later you "introduce" him as some pale creep Naruto doesn't know. Next how on earth could Kurama possibly know about Kohta's brief childhood connection to specifically Lucy?
Other details that don't sit right: Naruto and Lucy apparently not hanging out much until their first date despite the whole traumatic ordeal surrounding their affectionate attachment to each other. Lucy wanting to go by the name her captors gave her. Some of the people Naruto knows being so surprised at him even considering dating when he used to be accused of being a perv cuz he liked pretty girls and made his sexy justu from looking at porn and secondly he used to try to ask Sakura out on dates frequently to which she would LOUDLY reject him each time. Mariko's name being known without her being introduced verbally. Lucy not saying or even reacting to how awfully Sakura is treating Naruto her once only friend now best friend and saviour turned boyfriend and potential husband. Kohta deciding he wants revenge on a person for deaths of family he already came to terms with cuz some random people showed up and told him things that could be a lie and apparently not only discarding his nice life for this vengeance but also uprooting the lives of his cousin and the homeless girl they took in over it while apparently having no issue with the appalling methods such as cloning people for weapons that these people use which he should be against. The jonin chosen to train the girls freaking out more over the ice kekkai genkai than the wood one that only their previous Hokage was known for having naturally. Lucy thinking far less of Kohta/calling him a brat because he told a lie to her. Sakura after rudely and illegally breaking into Naruto's home like she has the right having the nerve to act like she has a conscious when it comes to him because she thinks that she'll feel guilty about assaulting him when he just doesn't want his daughter and future wife disturbed by her antics instead of defending himself(as he's entitled to and should do) against her unjustified bullshit anger.
That's all I can think of currently. Otherwise this was nice.

Where to begin? I mean, I loved this Fanfic entirely! (Even though I barely know anything about either anime (Elfen Lied it Naruto)) But it made me like it a lot. Keep doing it!

This is just really interesting and I can't wait for more

It should have been "Horny Rescue Team" instead of "Horned Rescue Team" in my opinion.

that was good keep the chapters coming

It's a great fic. Hope one day u finish it.

I felt like killing something after reading that last bit.

I can't think of any jutsu, but I have an idea. Using their vectors, can't they use multiple jutsus at once? Depending on the jutsu of course.

Hinata would had been with naruto too hell he should had dated both lucy and Hinata

I think I have a new idea for a fanfic

Oy, Blade! Are you ever going to continue these stories? I really enjoy reading them, and would like to see what happens in them.

Y

I believe Sai was already in the previous chapter or two...

This revelation to Orochimaru is like getting a gift for christmas because it is a powerful killing machine, and Kouta who used to be a person that liked Lucy in the past wants revenge like Sasuke.

I do have one question Lucy is Naruto's childhood friend but will their friendship blossom into something more.