1) Homeopathy is neither scientific, nor derived from plausible
mechanisms.
Homeopathy is based on the belief that:
a) "like cures like" (taking something that makes you nauseous should cure nausea)
b) water molecules retain a "memory" of the solutes once in the water solution is diluted to a point where none of the solute should be reasonably expected to be present at all. Somehow, succussion, or that act of tapping the solution with a leather glove between dilutions gives the solution this memory. Usually, the dilutions are to such an extent that it is statistically unlikely that any of the original solute is even present in your sample of the end dilution. There is no scientific basis for water (or any molecule) to retain a "memory". Anyone who then attempts to invoke quantum physics or quantum entanglement (which typically gets brought up at this point) is hoping the opponent/audience doesn't understand quantum physics. Quantum physics does not allow for this "memory".
What this really boils down to is "wishful" or "magical" thinking. My personal opinion is that this is like voodoo: you take something that makes you sick, you dilute it until its gone and you project the phenomenon upon your beliefs of your illness. Its the equivalent of one of those "pin cushion dolls".
2) There remains no large scale randomized, placebo controlled trials
that demonstrate any benefit beyond that of the placebo used to control
the trials. Any positive results in the past occurred in heavily flawed
and suspect trials that have never been able to be replicated when the
flaws were corrected.
3) This third statement basically amounts to an "appeal to popular
opinion". This demonstrates that homeopathy is popular, but it would be
a logical fallacy to leap to the conclusion that popular = effective.
Smoking is popular, crystal meth is popular, McDonald's is popular.
4) This statement is also a logical fallacy. The problems with the
"appeal to celebrity" are similar those associated with popular opinion.
Cheers