S&P500 held fast to their rules on consecutive quarters of profitability and forced TSLA to meet them (must be profitable in qX + sum to net profit over the past year). If they hold to them this time, SpaceX would need to be profitable over a year while public to enter the index.

They have instituted rules and gone back on them eventually (most notably for several years they had a "no going public with different classes of voting shares designed to allow control forever, if IPO is after today" rule that they eventually dropped) but they are generally pretty good about following rules.

I am not an expert, but my understanding is most funds don't change allocations immediately, but it would be part of normal rebalancing, e.g. VOO and other indexes that track the S&P500 do it quarterly

They all smear the purchases and sales from index changes, but I don't think they publish on what timescale. Most funds try to minimize tracking error. There are funds that take this to a different level. When a stock is added to the big indexes, it tends to do poorly over the next year, and on the flip side when a stock is removed it tends to perform well. Dimensional funds have automatic rules to take advantage of this type of thing. There are other companies that have funds of this style, but overall they are much less widely used than the big index funds from vanguard, blackrock, state street, etc.