Hi Aleksandr,
I agree with you that processes do not occur in the past. Instead of past time realms we have memories, records and artifacts of formerly existing configurations of matter and happenings.
You need to be clear what you mean by 'universe'. You are not referring to what is observed. That being space-time observation products So this is another version, an exiting universe not observed. Be clear what you mean by Present is it the seen content of observations currently produced (general parlance ) or when your unseen universe exists. Does time create thermodynamics or is thermodynamics and other movement processes perceived by us (via our senses) as time passing. I know what I think. Biological processes can act against thermodynamics but still progress in the 'direction' of the arrow of time.