Are there any spell level restrictions, above 3rd, for an Arcane Trickster Rogue without multi-classing?

If I'm understanding your question correctly, your confusion comes from the fact that they don't explicitly state whether they are referring to spell level or character level when they mention "level". If I've missed the point, then this answer is irrelevant (although still potentially useful for someone else who has the confusion that I've assumed).

If that is the case, we can determine whether they meant spell level or character level judging by the context of each occurrence of "level" (or a level like "3rd", etc).

For your first quote, some of it is explicit, but others not. I'm going to walk through the quote with [commentary in italics in square brackets]:

You know three 1st-level wizard spells [this is clear, they explicitly mention that they are referring to 1st level spells, so this is spell level] of your choice, two of which you must choose from the enchantment and illusion spells on the wizard spell list. The Spells Known column of the Arcane Trickster Spell casting table shows when you learn more wizard spells of 1st level or higher [again, it explicitly says spells of 1st level, so we're referring to spell level again]. Each of these spells must be an enchantment or illusion spell of your choice, and must be of a level [in this case, we must look to earlier in the sentence for context, which I've also highlighted in bold, to see that in this case, we are talking about spell level] for which you have spell slots. For instance, when you reach 7th level in this class [since they say "in this class", this time level refers to character level], you can learn one new spell of 1st or 2nd level [here we're back to spell level, as it is mentioned explicitly].

For your second quote, let's walk through it again:

Whenever you gain a level in this class [this is referring to character level, since they don't mention spells at this point but do mention the class], you can replace one of the wizard spells you know with another spell of your choice from the wizard spell list. The new spell must be of a level [the context of this occurrence of level refers to spell level] for which you have spell slots, and it must be an enchantment or illusion spell, unless you’re replacing the spell you gained at 3rd, 8th, 14th, or 20th level [this is potentially the most ambiguous part, but this is referring to character level, specifically because at these character levels, that's when you gain the "from any school" spell] from any school of magic.

Hopefully, assuming I've interpreted your question correctly, this resolves any ambiguity around what they meant by "level", etc., regarding whether they meant character level or spell level.

As for the answer to your question at the end of your post, "What is the highest spell level I can choose at each level for a Rogue Arcane Trickster?", it is as MikeQ's answer describes:

Assuming a single-classed rogue, this means the following:

• From rogue level 3 to 6, the Arcane Trickster has 1st level spell slots only.
• At rogue level 7, they gain 2nd level spell slots, so they can learn 1st or 2nd level spells.
• At rogue level 13, they gain 3rd level spell slots, so they can learn 1st-3rd level spells.
• At rogue level 19, they gain 4th level spell slots, so they can learn 1st-4th level spells.

The Arcane Trickster Spellcasting table specifies the highest spell level available for each rogue class level.

The Spellcasting feature in the Arcane Trickster subclass describes which spells the Arcane Trickster can learn. The level of spell is limited by the highest level of spell slot they have access to. For a single-classed rogue, this should match the spell slot progression in the Arcane Trickster Spellcasting table.

The relevant parts have been emphasized below:

The Spells Known column of the Arcane Trickster Spellcasting table shows when you learn more wizard spells of 1st level or higher. Each of these spells must be an enchantment or illusion spell of your choice, and must be of a level for which you have spell slots.

Whenever you gain a level in this class, you can replace one of the wizard spells you know with another spell of your choice from the wizard spell list. The new spell must be of a level for which you have spell slots, and it must be an enchantment or illusion spell, unless you’re replacing the spell you gained at 3rd, 8th, 14th, or 20th level from any school of magic.

Assuming a single-classed rogue, this means the following:

• From rogue level 3 to 6, the Arcane Trickster has 1st level spell slots only.
• At rogue level 7, they gain 2nd level spell slots, so they can learn 1st or 2nd level spells.
• At rogue level 13, they gain 3rd level spell slots, so they can learn 1st-3rd level spells.
• At rogue level 19, they gain 4th level spell slots, so they can learn 1st-4th level spells.