Can (or has) the Sanhedrin be (been) halachically established today?
If those whose answer imply that the Sanhedrin does not yet exist, then what are we to think of the so-called Sanhedrin that claim to now exist in Jerusalem?
There are those (like the Radbaz) who say that either Eliyahu HaNavi or the Mashiach himself will be able to re-establish the chain of semicha. Until then, it cannot be done.
Rambam says that if every Rabbi agrees on a person who is worthy of receiving s'micha, then that person can revive the chain. Once that is done, and he can identify 70 others who are worthy of s'micha, the Sanhedrin can be revived. However, so far, the practical difficulties have proven to be too great to overcome whenever such an attempt was made.
The current attempt (2004) has failed because the gedolim of this generation have not accepted the attempt.
One could say that if there were someone who was accepted by everyone as worthy of semicha, the mashiach would have come already (two Jews, three opinions).
If there was only one judge in Eretz Yisrael who possessed semichah, he should call two other judges to sit with him and they should convey semichah on 70 judges at one time or one after the other. Afterwards, he and these 70 should join together to make up the Supreme Sanhedrin and grant semichah to others to make up other courts.
It appears to me that if all the all the wise men in Eretz Yisrael agree to appoint judges and convey semichah upon them, the semichah is binding and these judges may adjudicate cases involving financial penalties and convey semichah upon others.
If so, why did the Sages suffer anguish over the institution of semichah, so that the judgment of cases involving financial penalties would not be nullified among the Jewish people? Because the Jewish people were dispersed, and it is impossible that all could agree. If, by contrast, there was a person who had received semichah from a person who had received semichah, he does not require the consent of all others. Instead, he may adjudicate cases involving financial penalties for everyone, for he received semichah from a court.
The question whether semichah can be renewed requires resolution.
Rabbi Kaganoff goes into detail about this dispute and explains what happened in the latest controversy.
His comment about the attempt in 2005 was
Obviously, this system carries absolutely no halachic validity according to any opinion.
Let's get back to the theory:
Isaiah Ch. 1 promises that in the future, God will bring back "our judges like in the olden days"; so we believe that at some point, we will have "real" ordination -- a prerequisite for a real Sanhedrin -- once again.
So how do we get that back if it is lost?
A.) Maimonides' position -- "all the scholars of the Land of Israel agree on someone." B.) Some variant on Maimonides' interpretation -- does it mean "all", does it mean "most" ... questions of wording. (When this was tried in Safed in the 1500s without consulting Jerusalem, the scholars there raised these questions.) C.) Elijah the Prophet is on the list of people who have "real" ordination, so when he emerges to herald Moshiach's coming, he can confer it then on others. D.) We don't know ... or we're waiting for God to figure this one out for us ... or it's something else.
Basically, the new Sanhedrin relies on A. (where Rabbi Halberstam was its nexus). If someone disagrees, they can easily claim B., just as the rabbis of Jerusalem did in the 1500s. (Though this time, they tried hard to consult all over Israel, unlike back then when they didn't bother asking Jerusalem.) C. is also certainly defensible, and some may just fall back on D. For any of these approaches, one could say about the contemporary Sanhedrin: "they might be of full halachic strength, I'm not sure"; or "they're well-meaning scholarly individuals but I disagree with their halachic logic on this point"; or something less polite than that.
The point is that this really centers around a halachic question with multiple suggested answers.
In answer to one of your questions,
Has the Sanhedrin been halachically established today?
The answer is that there have been some modern revival attempts, although all times they were far from being a universally accepted thing among all Jews and were oftentimes short lived.
Please see this Wikipedia page for further information.
This of course, remains a partial answer since you mention the halachic grounds as to whether or not the Sanhedrin can be revived in this day and age (without the Mashiach, Beit HaMikdash, etc.)