Why do some people have the custom to say, in the blessing on the Chanukah lights, the word "Shel" before "Chanukah" and some do not? And I have been told some people put the words together as "SHELCHANUKAH": why is that?
As with most issues of nusach, the proper one for you is whichever one your parents or teachers taught you.
Shulchan Aruch (Orach Chaim 676:1) has it without של, but Mishnah Berurah there comments that the Gemara and Poskim include this word, and Maharshal argues that the correct version is שלחנוכה (as one word).
There is a good summary of the various opinions, with sources, in Dayan Raskin's notes on the Siddur of R' Shneur Zalman of Liadi, here (note 8).
Magen Avrohom siman 676 says SHELCHANKAH (no space)
There is a dispute as to whether the bracha ends L'HADLIK NER SHEL CHANUKA or L'HADLIK NER CHANUKA. One should follow his own (or family) minhag, if you have one. If not, ask your Rav which wording you should use. (A third opinion is to combine the words with L'HADLIK NER SHE-L'CHANUKA.)
The following comments are not meant to encourage you to go with the NER CHANUKA version, but they might help explain why some people embraced this text for the bracha. (By the way, Rinat Yisrael siddur has NER CHANUKA. So does the GR"A siddur, Eizor Eliyahu. Koren siddur has NER SHEL CHANUKA. So too T'filat Kol Peh.)
Candles for Shabbat are candles, in the sense that they produce light (which is their main function), heat, fire - just like "regular" candles. Therefore, the appropriate term for them is NER SHEL SHABBAT, candles of or for Shabbat. So too, SHEL YOM TOV and SHEL YOM HAKIPURIM.
Chanuka candles are not "regular" candles. We are not allowed to use them - not their light nor heat nor fire. As we say, ELAH LIR-OTAM BILAVAD, (they are) exclusively to look at them, to see them. That's it. They are not candles that are being used for Chanuka. They're CHANUKA CANDLES.
[Here's another MASHAL... It's like the difference between the flowers one buys for his Shabbat table - here and outside Israel. In most other places of the world, one would go to a flower shop and buy flowers, using the flowers to honor Shabbat. They are flowers for Shabbat. But in Israel, one usually buys Shabbat flowers. They are flowers that are being sold for Shabbat by vendors who only sell flowers on Erev Shabbat and Chag. Same flowers, but different.]
Again, this is not meant to discredit referring to them in the bracha as NER SHEL CHANUKA. The gemara and many authorities since, have given the text as NER SHEL CHANUKA. The above was just meant to explain why other sources go for NER CHANUKA as the ending of the bracha.
Additionally, with NER CHANUKA, the first bracha has 13 words, corresponding to the 13 MIDOT of G-d's mercy. With the second bracha also having 13 words, the two brachot add to 26 words, 26 being the numeric value of HaShem's name. (NER CHANUKA and NER SHEL'CHANUKA both fit this numeric observation.)
When it is written in a single word, it should be read shelachanukkah. See any vocalized siddur that follows this custom. They have a dagesh in the lamed, due to the she, with a patach, an incorporated definite article. See also Shir hashirim 3,7 for this early stage of the genitive construction, preceding the emergence of the independent preposition shel, also attested in various manuscripts of the Mishna. See also she-la-yom ha-shevii in various nusachim of the first blessing before the Shema on Shabbat morning.