Yes, but only 1st-level spells
PurpleVermont has already argued why such a multiclass could only write down spells of a level they could cast as a Wizard (so actually up to 2nd level, if they take Wizard 3). I disagree with Vermont's argument that stolen spells could not be transcribed at all, however.
The rules don't clearly state that you have to have seen a spell written down to be able to transcribe it, that's a matter of interpretation--and I disagree with the interpretation:
Copying a spell into your spellbook involves reproducing the basic
form of the spell, then deciphering the unique system of notation used
by the wizard who wrote it. You must practice the spell until you
understand the sounds or gestures required, then transcribe it into
your spellbook using your own notation.
The fact that different wizards use different notation implies that the written form of the spell is not an inherent part of the magic, merely an aide-memoire. And the requirement to practice the spell until you understand its V/S requirements -- precisely the thing which the Arcane Trickster can instantly intuit -- strongly suggests that this is the most important part of "learning" a spell, and the rest is merely paperwork that your first-level Wizard training would absolutely prepare you for.
No, you don't have to pay for spells you gain when leveling up
From the PHB, p. 114 sidebar:
Your Spellbook
The spells that you add to your spellbook as you gain levels reflect the arcane research you conduct on your own, as well as
intellectual breakthroughs you have had about the nature of the
multiverse. You might find other spells during your adventures. You
could discover a spell recorded on a scroll in an evil wizard's chest,
for example, or in a dusty tome in an ancient library.
Copying a Spell into the Book
When you find a wizard spell of 1st level or higher, you can add it to your spellbook if it is of a level for which you have spell slots
and if you can spare the time to decipher and copy it.
As you see, the rules establish a clear difference between spells that add to your spellbook and those that you find and copy into it. If you take a look at the next part, you will know why.
Copying a spell into your spellbook involves reproducing the basic
form of the spell, then deciphering the unique system of notation used
by the wizard who wrote it. You must practice the spell until you
understand the sounds or gestures required, then transcribe it into
your spellbook using your own notation.
For each level of the spell, the process takes 2 hours and costs 50
gp. The cost represents material components you expend as you
experiment with the spell to master it, as well as the fine inks you
need to record it. Once you have spent this time and money, you can
prepare the spell just like your other spells.
As you see, the process of copying involves a lot of time, effort and materials which are dependent on you not yet knowing the spell and reproducing it yourself. For spells which you gain as you level, you skip all of this. The final sentence in the section makes it clear that once the spell is copied through this process, you can prepare it just like your other spells. This implies you do not go through this process for your other spells and that this is a different method of adding a spell to your spellbook.
Lastly, on the same page you are given these simple instructions on what happens when you gain new spells and when you do so.
Learning Spells of 1st Level and Higher
Each time you gain a wizard level, you can add two wizard spells of your choice to your spellbook.
And there is nothing more to it (aside from level restrictions on the spells).
Best Answer
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.