This is really something that could be googled with ease. Every source that computes GDP provides a very comprehensive explanation of the methodology, the release dates etc. Granted, these documents are usually very comprehensive and complex.
1 ) No. The reason we do not have monthly GDP is that it is too difficult to obtain data for higher frequencies. For example, even big public companies only provide quarterly and annual reports. Small firms frequently only provide one financial statement per year for tax reasons.
2 ) It depends what you look at (or what is made available). Usually GDP is simply the productive capacity (all goods and services - more on that below). In the link above, you see that the BEA annualizes quarterly GDP. So you cannot simply add it (even if it is the same method, non seasonally adjusted and all else equal). You can try this with GDP and GDPA from FRED (source of GDP is BEA). Taking the average of the 4 quarters within a year will do the job though (in this case).
3 ) and 4 ) see first link plus this one
That said, GDP is a always a guess. There is no way to know what the entire economy is doing for sure. For example:
- Nigeria doubled its GDP virtually "overnight" by changing how it computes it.
- How do yo handle illegal transactions? I lived and worked in London for several years. You must be blind to not see a sex worker or drug dealer ever other day. Ignoring them all, will grossly understate the true GDP. It does not matter if it is legal or illegal, it is still a service that is provided and consumed.
If you think I am nuts, google how the UK (and many other countries) compute(s) GDP. Hint, illegal drugs and prostitution are taken into account (to some extent). Funny enough, the UK only uses London and extrapolates it all out. I spent some time a few years back to "investigate" this because I personally thought I see more hookers in London than outside the city (realizing through the process that "street sex workers" are not counted anyway). If you can trust AdultWork, it seems my personal view was biased, and there are in fact plenty of hookers outside of London too.
Now, using hooker in the UK is generally a bit of a misnomer, because most people would associate this with the guy in the middle of the front row of the scrum, who tries to hook the ball (Rugby). That shows that definitions matter a lot. The UK excludes male and transgender sex workers, as well as street workers. In essence, whoever designed this, thinks prostitution is a female only, urban-centric and spatially immobile profession.
@Henry made a valuable comment. However, the monthly estimates cannot simply be added to quarterly values either. If you look at "How we disseminate the data" in Methods used to produce gross domestic product (GDP) data, you will see that around 80% data content for the output measure, 60% data content for the expenditure measure and 40% data content for the income measure are used for the first quarterly GDP estimate. The monthly data excludes the last two completely. The ONS provides some details explaining why the two do not provide the same estimate as to where the economy is relative to its pre-pandemic levels (which is a good guide to understand why you cannot simply add months in reality).
If you quickly search monthly on the website, you see that monthly data is currently using 2019 = 100, which means that the average of all months in 2019 equals 100. So quarterly GDP, even if there were no revisions or data uncertainty, is not simply the sum of 3 months because of the way the data is provided.