Is there a difference in meaning between Oglądać (Obejrzeć - perf) and Spoglądać (Spojrzeć - perf)?
Yes, a significant one. "spojrzeć" is just to take a look, it's a very short action. "obejrzeć" is used when you watch sth thoroughly, that is why it's for example used with movies: "obejrzeć film" (to watch the whole movie). You could say "Obejrzałem ją", but that would be something like "I examined her" (it may possibly be something that a doctor says).
The closest to 'checking out' is "obczajać" (perfective "obczaić"), but that's clearly quite colloquial ;) For less colloquial speech I guess "spojrzeć" (imperfective "spoglądać") works well.
Why is it "nią" here (Instrumental)? I thought it was either Locative or Accusative after "na" was used.
Thanks. I was using https://en.wiktionary.org/wiki/Appendix:Polish_pronouns as my reference and when you expand 3rd Person Pronouns it only shows nią against instrumental. That's twice I have found the English Wiktionary to be wrong compared to the Polish version.
I've been using this link too and it shows the correct declension for sg. fem. accusative now: "ją/nią". Maybe they've updated.
For a short glance the best option is "Zerknąłem/Zerknęłam", but close enough, added.
What exactly is the difference between "patrzyć" and "patrzeć?" I've entered the past tense form of "patrzeć" before but it marked it as wrong because I didn't use the past tense form of "patrzyć."
They are both correct and mean exactly the same, but "patrzyć" is 20 times less common (well, that's 2001 data) and even Polish people may wonder if it's actually correct.
Interestingly, while "patrzeć" is more common than "patrzyć", the past forms of "patrzeć" (patrzał, patrzała) are a lot less common than those of "patrzyć" (patrzył, patrzyła).
When i read the *«spojrz» part i just knew it was "look" somehoe by singing the „Sofía” song... "Bez twój spojrzenie, jestem bez twój spojrzenie" :D