Should I include 'out of stock' items in the Google Shopping Feed?
Google generally recommends to include 'out of stock' items in the feed for products that are temporarily unavailable. By doing this, it eliminates the potential for an activation delay due to a review when a product is re-introduced.
In practice, a short-term (less than 30 days) removal of an item does not incur any review delay when re-introduced. A longer term (more than 30 days) removal of an item also typically incurs no delay when reintroduced. However, especially in certain categories, there could be a delay of anywhere from a few hours to a few days. This is not documented anywhere, just based on experience.
However, 'out of stock' items do NOT appear in Google Shopping search results, and do not receive any traffic from there.
Product feeds do not have any effect on Organic Search listings, so those would continue to appear in Organic Search with price/availability using the structured data tags on your site.
In cases where the catalog has many errors/warnings, it is preferable to filter the 'out of stock' items from the feed. The reason for this is that you can then see the errors and warnings only for actively listed items in the diagnostics, and then work on correcting those errors. Otherwise, those get drowned out in all the other errors on 'out of stock' items.
It is very important to get the errors on 'in stock' items fixed and data quality increased (e.g. by adding images), because those are the only items that are eligible to appear in Google Shopping search results.
If your items are available on a delayed basis, such as preorder, call-to-order, drop-ship, or backorder, then you can use the 'preorder' or 'backorder' availability values. Google requires an 'availability_date' with these values. If you do not have availability dates, then the feed can be programmed to set 'availability_date' to a date that is two weeks out (for example).
Note that 'preorder' is for items which have never been on sale before, and 'backorder' should be used for all other cases of delayed availability.
For further details, see here: Availability [availability] - Google Shopping Feed Specifications