Let me start by explaining what I think of as "everyday magic". I am talking about the low level, low cost spells that can be cast by almost anyone. For Mer, these could be quite powerful, since they seem to be a more magical race with a higher level of natural magika. Even for Men, these little cantrips could be quite useful, and maybe even provide an alternative to more technological methods. For instance, instead of matches, what if everyone could cast a mini-fireball to start campfires in raging blizzards. Instead of relying on advances in medicine, nearly everyone could heal scratches and scraqes, without fear of infection. Tamriel seems to be a very "magical" land, and nearly everyone seems to have some magical ability. A random thief in the Imperial city dungeons can cast a fireball strong enough to kill a rat. Even in Skyrim, where magic use is frowned upon, an escaping prisoner knows some healing magic.

For gameplay, we we are limited to the big, impressive, world shaking spells, but what else could magic be used for, and how would it affect a culture? For example, "summon almost-invisible friend" would make many children's game more fun, and "static discharge" could be used to chase small animals out of the house. "Change hair-color" would be very useful for that party at the castle. Or do you think magic is more limited and restricted for some reason?

I'm pretty sure starting off with magic no matter who you are is a game-play thing; Not every citizen actually knows magic, probably more people who don't than do.

I know from a lore-perspective, it takes years of training just to master a pretty basic spell.

This everyday magic as you put it is something I have the most trouble getting used to. I tend to play characters that either do not like using magic or are mages, although it doesn't make much sense. After all, Tamriel's people can naturally use magic, which doesn't mean that they will want to enter the College of Winterhold; I'll have to rethink my characters.

It depends. In some Tamrielic cultures magic use is so wide-spread that magic itself becomes vulgar. Bretons and Altmer are probably good examples.

I'd say spellcasting is not so common. While every lowly peasant in Tamriel does have the capacity for spellcasting, it seems that the level of skill required for spellcasting is on par with learning a trade. We never hear about the lowly peasant casting cure disease to deal with his cold, for instance. Instead, we have quests about getting someone a cure disease potion from a temple priest, a well-established need for healers, and a lack of familiarity with magic among the common man. Magic is a bit like blacksmithing, mechanical engineering, or some other skill in that just about every healthy advlt has the physical capacity for the work (they've got hands, and a brain) but you need to be trained and the majority of people don't have that training.