I have found some affordable tickets on BudgetAir (could be any other service - specifying it just for the reference in case someone needs it) and I wonder if I can choose the seats from the carrier (Kuwait Airways)'s own page later on? I know they allow it for the normal tickets purchased from their website, but do they facilitate it even for the 3rd party ticket purchases too? Issue with BudgetAir is that its not showing some of the seats in seat selection which are available on main KWI Airways page.
There is no general answer for all airlines and all booking sites.
Some airlines will allow people who bought from an other seller all the same perks as people who bought from the airline itself.
Other airlines restrict people who bought from other sellers in some or many ways.
Airlines do not have to publish what rules they follow, when they change them nor whether they keep the same rules for all resellers.
So even when you know about the rules for the combination of airline and reseller you can run into problems when the airline changes those rules.
A good reason to avoid resellers is that they have often proven difficult in getting your money back when the airline cancels the flight, in getting you your money if you get a delay and the airline is bound to compensate. And often they sell you 'a ticket' for two flights which are unrelated for the airline(s) and leave you hanging when one of them is delayed or cancelled.
And one small thing I noticed myself, the airline send an e-mail to announce a change in departure time and the reseller took 24 hours to send on the mail. In my case it was a departure a bit later which did not matter anything, but it could have cost me big if I had had two different flights or if the change had been bigger. For me the reason to avoid resellers since.