tę can be translated as both this and that, though it's kinda weird that in some cases they don't accept "that" but here it's put as the main translation
Same in Russian. "Это" (eto) means "this," but translates as either "this" or "that" in many cases in English.
Вижу эту женщину. (Wiżu etu źenśćinu.),
"I see that woman."
yeah, well I'm Russian so I already know that ;) anyway, it's kinda nice that in these languages you don't really have to worry about whether it's "this" or "that" (or "it"), like in Spanish or Portuguese for example, their words for "this" and "that" also differ in 1 letter only and you always have to watch out which one it is, as they don't accept both interchangeably
For me Russian "это" (this) is very different from "то" (that). I'd translate Вижу эту женщину as "I see this woman".
You're right, they are very different in Russian, and "то" is rarely used like the word "that" in English as in this exercise. "I see that woman," would still use эту женщину instead of "ту женщину," unless you are specifically contrasting "that woman" from "this woman." In that case you would say, for example, "Я вижу эту женщину и ту женщину тоже." "I see this woman and that woman too."
Why Widzę tę kobietę and not widzę tą kobietę? Does 'ta' take that form after a verb because it is a determiner and not an adjective?
ten/ta/to is not an adjective. It's a noun determiner when followed by a noun. In your case, tą is in the instrumental case. Kobietę is in the accusative case, so its noun determiner tę and must match its noun.
Seeing = dating, that's an English equation. In Polish, seeing is just with your eyes.