The answer to the former question is yes. If you have the Influenza vaccine it interferes with your immune system to respond to covid19. This data came from looking at military personal who were vaccinated in the 2017-2018 period.
What they found was that there was a 36% increase in your getting covid if you had the influenza vaccine. Click on the image below if you want to read the whole paper. To a theoretical physicist like me, this result is quite obvious, because if the immune system is primed to look for the influenza virus, the others might sneak past. Of course, sometimes, you have to do the experiment as a theoretician knows. The exception is a Gedankenexperiment when a theoretician does it in his head like Galileo Galilei (leaning tower of Pisa) and Albert Einstein (for both special and general relativity) did.
They called this viral interference.
The question now is, if you are vaccinated against covid19, will that increase your chances of getting the influenza virus. My head says yes, as your system will be over primed to look for the spike protein of the coronavirus. This is why Omicron is so illusive. It has less spikes and the spikes are different to the ones used in the vaccine as well. Incidentally the spike protein on the SARS-C0V2 coronavirus was patented by Moderna in 2017, before the pandemic stated. Yet another nail in coffin that this came from animals and was not made in a lab.
We should have some data from Europe and the USA and it is probably being analysed right now, but we will soon find out in Australia.
What if you have both jabs? I would think some virus is going to like the path of least resistance you will create for it by focusing on Influenza and Covid. Viruses are very opportunistic.
The the games/experiments begin.
I say a similar graph released by Queensland health. The Flu season is off to a flyer because it has not been in Australia for the last 2 years. We will also see how much the the Covid jab affects people’s ability to handle the common cold/flu this winter. I expect there will be major issues.