When choosing 2 new spells for a high INT Wizard achieving 2nd level, can they select 2x 2nd level spells?

I'm a little confused due to everything I'm reading online that explains the process missing this one clarification.

Every level, a wizard can add two new spells to their spell book so long as they're capable of casting the spell. Does this check of capability work against the base wizard spells per day table only, or does it also include the bonus spells based on high INT?

So my 18 INT Wizard gets 1x bonus 4th level spell per day. Does this mean that at 2nd level they can choose a 4th level spell?

To get bonus spells of a given level, you must already be able to cast spells of that level. Since a 2nd-level wizard can cast spells that are at most 1st level, you only get bonus 1st-level spells, regardless of your Intelligence. So your 2nd-level wizard with 18 Intelligence gets one bonus 1st-level spell, and no bonus 2nd-, 3rd-, or 4th-level spells, no matter what the table says.

Since you don’t have those slots, you aren’t capable of casting those spells, and thus cannot choose them as your freely-scribed spells. You’ll have to choose 1st-level spells for that.

You can, however, scribe higher-level spells if you pay for them. The process of paying to add for spells to your spellbook beyond the free ones doesn’t specify that you have to be able to cast them. So if you come across a higher-level spell, and you’re able to understand it well enough to add it to your spellbook, you can do that, so it will be ready when you get high enough in level to cast it. But you still need to get to that level for it to be useful to you.

Ultimately, anything that allowed a character to cast a spell that is higher level than is listed for a wizard of their level in Table 3—18: The Wizard is utterly game-breaking. Just the spells the wizard usually gets is often game-breaking, but going beyond that is definitely a bad idea.

For the record, spells you pay to add to your spellbook don’t even have to be wizard spells, technically. You could, if you wanted, write down a cleric spell or whatever in your spellbook. There is usually little point to it since they do have to be wizard spells to prepare and cast them, so it’s pretty much just a waste of time, ink, and paper. But maybe you would if it’s a plot point or something.