In this situation, most of the decisions would likely be reversed by the Supreme Court. How swiftly could they reverse? Would the increased number of disputed cases slow down the Supreme Court's normal functioning?
I am unaware of any factors that would prevent the Supreme Court from reversing decisions as quickly as it wanted to in this situation. There are the so-called "shadow docket" cases in which the Supreme Court issues an order with reduced briefing and argument, and often with little explanation of the decision.
In addition, the court may choose to "consolidate" multiple cases that involve the same legal question.
Would the judges face reprimand or the threat of removal?
As far as I can tell, it is not normal to issue a reprimand simply because a judge issued a ruling that a higher court considers to not follow precedent.
Likewise, removal of a federal judge is generally considered to be an exclusive prerogative of Congress, by impeachment, and in practice impeachment is the only way federal judges get removed from office. Some legal scholars have argued that it might be Constitutional to establish some other means by statute (I linked to this argument in this prior answer), but I think this is purely theoretical.
What incentives prevent this from happening right now? As I understand it, most lower court decisions treat Supreme Court precedent, especially recent precedent, as inarguable truth, forming the axiomatic basis of the lower court's decisions. However, when lower court judges disagree with the recent Supreme Court decisions or philosophical trends, what stops them from ruling based on their own judicial philosophies instead?
I found a paper that discusses some of the incentives that lower courts have to avoid reversal, and tentatively argues based on statistics that we do in fact see courts responding to a reversal of their decisions:
"Patterns and Consequences of Judicial Reversals: Theoretical Considerations and Data from a District Court", by Joseph L. Smith, The Justice System Journal, Vol. 27, Number 1 (2006)
Smith mentions:
judges may fear that being reversed will negatively affect their reputation in the legal community (p. 3.)
they may be concerned that having a record of being frequently reversed will be counted against them when they have an opportunity for promotion (p. 3-4)
if the judge's initial decision was intended to advance a judicial doctrine or legal policy, a reversal represents an ultimate defeat of that goal, and thus may amount to 'wasted' work for the judge, and a potentially reduced ability for the judge to influence the long-term development of the law (p. 4)