Translation:I want to send them beautiful flowers.
Close. Quiero que requires subjunctive because it's sort of like a wish. (So you got the morphology--the form of the verb, correct, manden, but its mood is subjunctive rather than imperative.) If we wanted to add "to them" (to slightly alter the sentence), I think we would need "les" but "a ellos" would be optional: Quiero que ellos les manden flores hermosas a ellos, "I want them to send beautiful flowers to them." But I think we could use Quisiera, "I would like," in which case we are back to using the infinitive: Quisiera mandarles flores, " I would like to send them flowers." I'm not sure which form is most natural to Spanish, however, as I am not a native speaker. When I did this one (Yo les quiero mandar flores hermosas) I got it wrong--I put "I want them to send pretty flowers"--so I appreciate the question and the response here.
Yes, because "ayudar" requires a preposition to link it to an infinitive. "Querer" does not.
Unfortunately, you have to learn these individually; there's no blanket rule to tell you which verbs need a preposition and which don't. But if you pay attention to the first verb, and note its preposition (or lack thereof), over time it'll start to "sound right."
But one is more so than the other, and it all depends where you're from and what you are talking about. Then you have to fit in "lovely" ! I haven't been able to work out what the gradation is in Spanish yet, but DL has bonito, hermoso and bello (so far), and is inconsistent about what it accepts as a translation.