How are investments distributed across an index?

Basically it depends on the fund. For every fund there is a prospectus that states what the fund is doing, how it will do it and what this will cost your. It is strongly recommended to read this before buying a fund.

An index fund typically has a rebalancing interval stated in its papers, often quarterly. This means that it will 4 times a year sell stocks that are no longer in the index and use the money to invest in the stocks that joined the index. Passive management does not mean that there is no management, it just means that there is little management and management is not directed towards picking certain stocks.

How stocks in an index fund are weighted depends on the underlying index. Most indexes are weighted by market capitalization which means more valuable companies get more weight. The S&P 500 is one of those. This has the great advantage that appreciation of a certain stock will not mess up your weighting as a company that doubles its price will also double its absolute weight in the index (although in reality other stocks will increase their absolute weight as well, so relative weight will not double).

So first let's clear up some terminology:

An index in not a "fund". It is just an abstract list of companies and their relative weights to make up some number. You can't invest in an index directly. Instead, you buy units of an index fund that tries to replicate the returns (changes) of that index, either by fully replicating if they have enough capital, or by sampling, and adjusting periodically to more closely match the index.

ETFs are just mutual funds that are exchange traded. Not all index funds are ETFs, and not all ETFs are index funds. The main difference is that buying units of an ETF does not necessarily give more money to the fund. It might, or you may just be buying someone else's units and the fund is unchanged.

The allocation of funds within an ETF will be outlined in its prospectus. If it tracks an index, it will rebalance periodically (usually quarterly), and typically align with the indexes they track, to match the index as closely as possible. If it's an active fund, it will rebalance according to its investment objectives.

If companies enter and leave the index, the fund will buy ans sell shares in those companies (and other companies within the index) to try and replicate the index as best as possible.