Tenth walker
There's a reason, you know, why the Fellowship was composed of nine persons, Lord Elrond says it very clearly (FotR, The Council of Elrond):

The Company of the Ring shall be Nine; and the Nine Walkers shall be set against the Nine Riders that are evil.

If you want to add a person, you have two choices:
- leave a canon member of the fellowship out
- let Frodo become the tenth Ringwraith as a consequence of his wound (you have now two empty spots in the Fellowship).
Your choice.
But please, don't butcher canon adding a tenth walker without balancing the numbers again. An acceptable (in my opinion) escamotage would be to make your additional character join the Fellowship after they left Imladris: this way (s)he wouldn't be an official member and could travel with them without messing with the symbolism.

Westron as English
If you drop a modern person in Arda (s)he will likely have problem understanding people (and in being understood).
Westron, the Common Tongue among races of Middle-Earth at the end of the Third Age, has been "translated" in English by Tolkien in the books, but it's not English.
It's a totally different language derived from low-class Adûnaic (which was the tongue spoken in Númenor).
This also means that Tolkien "translated" all the Westron names in English: Bilbo Baggins is, in Westron, Bilba Labingi; Frodo Baggins is Maura Labingi, and so on. You can read more about it here.
Almost the same goes for the Rohirric and all the names of the Rohirrim (btw, "Rohirrim" is Sindarin, people of Rohan called themselves "Eorlingas"): Tolkien rendered this language with Old English, and basically didn't give us Rohirric words other than the element lô-/loh-, meaning "horse".

On Elves in general (and Legolas in particular)
Elves are one of the most common target for canon butchery. Having been portrayed by Orlando Bloom in the movies, Legolas in particular is an easy target for all kind of crazy teenager Mary Sue romances. There are several misconceptions about Elves in fanon (and debunking them would probably require an entire article), I'll summarise just the most important things, leaving links for further reading:

  • even before starting to research about Tolkien Elves you should remember that you're writing a character (at least) some centuries old. Here some tips on how to do it right

  • Legolas is son of Thranduil, who happens to be king of Mirkwood. Despite that, in canon, Legolas' social status and/or ranking isn't stressed in any way. Using "Your Highness" and similar phrases to address him in your fic is a bad idea. As it is creating some drama about him being "heir to the throne". Tinw wrote a detailed and extremely interesting analysis of the Legolas' rank issue

  • I enjoy here and there slash fics, but really, there's nothing in canon about a sexual relationship between Legolas and Gimli. If you really must do it, do it with taste and avoid the bane of all fan fictions: Mpreg (Male Pregnancy). Not only it's anatomically impossible but in this specific situation is also highly improbable, given that Gimli and Legolas belong to two different species. Yes, this applies also to Legolas/Aragorn slash.

  • and speaking of Legolas and Aragorn: there's nothing in canon to support the idea of a long-time friendship (before the War of the Ring) between them. The movies seemed to imply it, but Tolkien never wrote about it. They probably knew each other, as Aragorn handed over Gollum to the Mirkwood elves, but that's it.

  • and to further burst all your dreams of Legomance, I'd like to point out that in canon interspecies love is extremely rare, and exists only for Men/Elves relationships:

    There are four such recorded marriages in Tolkien canon: Beren and Lúthien, Idril and Tuor, Aragorn and Arwen, and Mithrellas and the first Lord of Dol Amroth. Mithrellas didn't stick around; she ran away from her husband (History of Galadriel and Celeborn, UF). Beren and Lúthien, and Aragorn and Arwen, wound up with the elf-ladies becoming mortal. This means that not only do they die instead of living an immortal Elf life, their souls leave the world and spend eternity with Men's souls. Idril and Tuor managed to scrape up a happy ending, sailing together into the West and being accepted there. (Silmarillion)
    (from What Tolkien Officially Said About Elf Sex by Tyellas)

  • Elves don't "fade from grief", (but they certainly die from it) nor they heal with magick glowing hands

  • finally, if you really must use Elvish names or phrases, do all of us a favour and do not invent them: there are some very good name lists and frasebooks out there. Don't use Grelvish.

Elladan and Elrohir as pranksters
One of the most inexplicable misconceptions around is about Elladan and Elrohir, sons of Elrond, who are often written as a sort of Middle Earth's Weasley twins.
Mackenzie W wrote a really good blogpost on "Fanon Beliefs Which Need to Die", where she addresses this problem:

Look, we all loved Fred and George Weasley. In Harry Potter. You don’t have to turn every set of twins into the Weasleys. [...] Tolkien describe Elrond’s sons as “somber.”

And somber they are, given that their mother, Celebrían, was so horribly tortured by the Orcs that even Elrond couldn't heal her and she sailed for Valinor, leaving her much loved husband and children in Middle Earth. After that Elrond's sons spent their time chasing Orcs.
Not playing pranks.

Childlike Hobbits
Let me borrow again Mackenzie's words here:

The Hobbits are child-sized, not child-like.

When the Quest starts, Frodo is fifty. While it's true that Hobbits age differently from Men (and they enters adulthood at 33), no one would have trust Frodo with the destruction of the Ring if he (and the others three Hobbits) had behaved like a child. When, during the journey, the Hobbits ask for songs they are not asking for lullabies: in a culture mainly based on oral tradition, songs were the preferred way to transmit history, tales and lore in general.