No, these aren't official rules from 5e, AFAIK
Nowhere in the Spellcasting chapters from PHB or in the many variant rules from DMG I'm aware of any ruling making you roll, for example, to be able to cast Cure Wounds.
Rulings like these would need to be playtested alot and carefully made, since this changes completely the balance of spellcasting.
Earlier editions and other systems had rulings about "rolling to see if you are able to cast a spell", specially when trying to cast spells while wearing armor.
In 5e, these conditions were made simpler: you can't cast spells if you are wearing an armor you are not proficient with or if you can't provide one of the components to the spell (V, S or M). Otherwise, you always succeed at casting the spell, unless the spell itself needs a check (Dispel Magic, Counterspell, etc). In particular, most support spells always succeed as long as the target is willing.
my DM is completely sure that he is following the rules step by step.
Ask him, before or after a session, where is this rule written. He can be reading some unofficial ruling or just misunderstanding the official ones.
This ruling benefits martial classes
Simple as that: if you are now making classes that didn't need to roll need a roll to succeed, classes that needed to roll because their actions would be mostly "Attack" anyway are now stronger (rather, spellcasters are now weaker).
There is alot of complaint in D&D about how spellcasters are better than martial classes in everything, but 5e has reduced this problem, and using an arbitrary rule without playtesting or careful thought usually isn't a good solution.
So, essentially, if you go against a party of full casters, you are happier. If you go against a party of barbarians, having spellcasters in your party, you are sad.
30 seconds rule
Welp, as already mentioned in comments and answers, ask your DM why is he doing it. Also, spells that last for 1 minute are supposed to last for 10 rounds. If each rounds now lasts 30 seconds, the spells should last for 5 minutes.
Repeating: randomly/arbitrarily changing the balance of the game like this usually is a bad idea, unless he actually thought alot in this change and preferably playtested it (or your current party is the playtesting table, in which case he should warn you about it, and handle it as a playtesting table, which means changing the rules on the run would be common when he notices it's not working).
Overall, your DM seems to be interested in nerfing Spellcasters
Ask him if that's the case and, if it is, why. He might think spellcasters are too powerful in the current state (which is a common opinion). Then, talk to him if these are the best changes to achieve whatever he wants, and ultimately express your concerns about how this is affecting (in a bad way) you and your character.
As I mentioned in the comments, these questions should be separated so we can actually address each one individually with more details.