You can hear Christopher Tolkien saying "Elanor" in a recording of him reading the final passage of The Lord of the Rings. He gives the three syllables about equal emphasis, but I think the the stress is on the first: EL-ahn-or.
This matches J. R. R. Tolkien's published account of stress in Sindarin words in Appendix E of The Lord of the Rings, where a three-syllable word is stressed on the second syllable if it contains a long vowel, diphthong, or cluster of multiple consonants; and otherwise on the first. The "a" of Elanor is none of those. This rule is not totally infallible, since there are several known exceptions (e.g. some readers feel Imladris was stressed on the "Im", despite the "dr", based on the metre of the "Seek for the sword that was broken" poem), and examples of Tolkien changing his mind. But this particular word seems to follow his principle -
Where (as often) the last syllable but one contains (as often) a short vowel followed by only one (or no) consonant, the stress falls on the syllable before it, the third from the end.
Analytically, the flower elanor is named after the two elements êl, a star, and anor, the sun. The long ê is lost in compounds (such as "Elrond"), so both the flower and the child have this sound as a short "eh" instead of a prolonged version.
Christopher Tolkien does not trill the final "r", notwithstanding his father's claim that the letter "represents a trilled r in all positions". This is a sound that is rolled at the front of the mouth, as opposed to being gargled at the back ("a sound which the Eldar found distasteful", used by Orcs and some Dwarves). But we can hear recordings of J. R. R. Tolkien himself giving varying degrees of trill to the "r" - that one is the poem "A! Elbereth Gilthoniel", where to my ear, the "r" in "silivren" is much more trilled than the one in "miriel", for example. So I think that the trill cannot be totally mandatory, or at least there is some consideration of euphony for when to sound it or not, even within the same utterance. Moreover, the amount and quality of trill seems to be a matter of accent within Middle-Earth, and it's not inconsistent for a Hobbit to receive the word, as a name, said in a less fancy way than Galadriel might have pronounced it.