First, what structure does your index fund have? If it is an open-end mutual fund, there are no bid/ask spread as the structure of this security is that it is priced once a day and transactions are done with that price. If it is an exchange-traded fund, then the question becomes how well are authorized participants taking advantage of the spread to make the fund track the index well? This is where you have to get into the Creation and Redemption unit construct of the exchange-traded fund where there are "in-kind" transactions done to either create new shares of the fund or redeem out shares of the fund. In either case, you are making some serious assumptions about the structure of the fund that don't make sense given how these are built.
Index funds have lower expense ratios and are thus cheaper than other mutual funds that may take on more costs. If you want suggested reading on this, look at the investing books of John C. Bogle who studied some of this rather extensively, in addition to being one of the first to create an index fund that became known as "Bogle's Folly," where a couple of key ones would be "Common Sense on Mutual Funds: New Imperatives for the Intelligent Investor" and "Bogle on Mutual Funds: New Perspectives for the Intelligent Investor."
In the case of an open-end fund, there has to be a portion of the fund in cash to handle transaction costs of running the fund as there are management fees to come from running the fund in addition to dividends from the stocks that have to be carefully re-invested and other matters that make this quite easy to note. Vanguard 500 Index Investor portfolio(VFINX) has .38% in cash as an example here where you could look at any open-end mutual fund's portfolio and notice that there may well be some in cash as part of how the fund is managed. It’s the Execution, Stupid would be one of a few articles that looks at the idea of "tracking error" or how well does an index fund actually track the index where it can be noted that in some cases, there can be a little bit of active management in the fund.
Just as a minor side note, when I lived in the US I did invest in index funds and found them to be a good investment. I'd still recommend them though I'd argue that while some want to see these as really simple investments, there can be details that make them quite interesting to my mind.
How is its price set then?
The price is computed by taking the sum value of all the assets of the fund minus the liabilities and divided by the number of outstanding shares. The price of the assets would include the closing price on the stock rather than a bid or ask, similar pricing for bonds held by the fund, derivatives and cash equivalents. Similarly, the liabilities would be costs a fund has to pay that may not have been paid yet such as management fees, brokerage costs, etc.
Is it a weighted average of all the underlying stock spreads, or does
it stand on its own and stems from the usual supply & demand laws ?
There isn't any spread used in determining the "Net Asset Value" for the fund. The fund prices are determined after the market is closed and so a closing price can be used for stocks.
The liabilities could include the costs to run the fund as part of the accounting in the fund, that most items have to come down to either being an asset, something with a positive value, or a liability, something with a negative value.
Something to consider also is the size of the fund. With over $7,000,000,000 in assets, a .01% amount is still $700,000 which is quite a large amount in some ways.