Can a low-level wizard learn and cast higher-level spells?

Question

Suppose you're starting with a level-1 Wizard with the universal arcane school and +5 in INT. The high modifier grants him extra slots, for example one 3rd-level spell slot. Suppose at the end of his first adventure the wizard find a scroll containing fireball. Can he learn the spell, writing it into his spellbook and cast it the next time, using the appropriate spell slot?

Furthermore, suppose the wizard reaches lv. 2: can he choose, as new spells to write on his spellbook, one 4th-level spell and one 5th-level spell (just to make an example), whose slots are again granted by the high Intelligence?

Answer 1

Can I use bonus slots to cast spells above what my class level normally allows?

No.

According to the official Paizo FAQ, you can only use bonus slots for spell levels that your class and level has access to.

For an example of this in action, the Paladin has a "0" in his Level 1 spells per day entry at level 4 (note that up to level 3, he has a "--"). This means he is granted 0 level 1 spells but can use his bonus slots, if he has any, to cast level 1 spells. As far as I am aware, only the ranger and paladin classes fall under this scenario, where bonus slots might grant casting before a class feature gives any per day spell slots.

Copying A Scroll into a Spellbook

There are no restrictions on spell level aside from the usual checks required to copy a spell into the spellbook. The wizard deciphers the magical writing on the scroll and once he has done so, he studies the spell for a minimum of one hour. At the end of the hour he makes a spellcraft check (DC 15 + spell level). If the check is successful, the spell is copied into the book regardless of the level. Copying the spell into the spellbook also costs GP based on the spell level; the exact costs are found in the table under the header Writing a New Spell into a Spellbook on the PRD page for Magic. If the spellcraft check fails he can't try again until a week has passed; the scroll is not consumed.

Answer 2

No.

The Bonus Spells slots for higher ability score is only additional spell slots once you reach a Wizard Level high enough to personally cast any spell of that level.

Absolutely, if you are a Level 1 Wizard, you can only prepare level 1 spells.

You cannot prepare any Level 2 Wizard Spells till you reach Wizard 3.

It is obviously absurd that a Level 1 Wizard with INT 20 (IntMod +5) can prepare and cast a level 5 spell. I don't think you would like a Level 1 enemy Wizard to have such capability against you.

"Suppose at the end of his first adventure the wizard find a scroll containing fireball."

While it seems a Wizard 1 can copy it into his spellbook by following all the appropriate checks, he cannot prepare it until he becomes a Level 5 Wizard as that is the level that a Level 3 spell becomes possible for Wizard to prepare.

A lower level wizard may directly cast a scroll of a higher caster level than his own, by making a caster level check:

1d20 + your caster level vs a DC of 1 + Scroll's caster level

If the Fireball Scroll was scribed at the minimum caster level of CL5 then you have an 85% chance of activating the scroll.

This of course will remove the spell from the scroll.

suppose the wizard reaches lv. 2: can he choose, as new spells to write on his spellbook, one 4th-level spell and one 5th-level spell

NO.

From the source that states how Wizard gets new spells:

"Each time a character attains a new wizard level, he gains two spells of his choice to add to his spellbook. The two free spells must be of spell levels he can cast."

Even if a Level 1 or Level 2 Wizard were to somehow end up with even a 2nd level Spell in their spellbook they could not prepare it and count not cast it.

Level 2 spells = Prepare From Wizard Level 3

Level 3 spells = Prepare From Wizard 5

Level 4 spells = Prepare From Wizard 7

Level 5 spells = Prepare From Wizard 9

Level 6 spells = Prepare From Wizard 11

Level 7 spells = Prepare From Wizard 13

Level 8 spells = Prepare From Wizard 15

Level 9 spells = Prepare From Wizard 17

There are no viable shortcuts to skipping all the vastly under-appreciated lower level spells. Not that fireball will help your overall successes anywhere near as much as a level 3 spell like Haste would.

See Sacred Geometry get more out of lower level spells.