I'm a native English speaker. It's quite alright to say "I don't see this dog" even if its not visible to you.
Maybe it's a dog we were talking about just now, so conceptually its "this" dog (and "that" dog would be equally acceptable). However lets say we were discussing two dogs, and we just finished talking about one of them - then to distinguish it from the other dog, that is further away conceptually and in time, then we would say "this dog (of which you speak)"... but the bit in brackets is optional, and just makes it clearer, and is not the only way to further clarify.
another person in the conversation can see "this dog", even though you can't (you're in different room, on the phone etc.) and has just mentioned it. So you could use "this dog". Especially if the other person has referred to it as "this dog" or has referred to a number of dogs and you want to specify one in particular.
hmmm probably not much difference between this example and the one above... more of a specific case of example1 ... anyway...
HTH (hope this helps)
If I remember correctly what immery always writes, Polish has "ten/ten/tamten" while English has "this/that/that". So the conceptions of... closeness are a bit different. That is why we try to accept both "this" and "that" in every translation of Polish "ten", but "tamten" is already so far that it can only be "that".
The gender of the pronoun doesn't pay any role in it. And yes, both your examples are in Genitive.
The basic translation of "this" is "ten", and of "that" is "tamten". Those forms are Nominative (so they mostly serve as part of the subject) and masculine. The feminine form is "ta" and neuter "to".
Now, "widzieć" (to see) needs Accusative. In Accusative, it is important whether a masculine noun is 'animate' or 'non-animate'. The animate version is "tego" and non-animate is "ten". Feminine is "tej" and neuter is, again, "to".
When a verb that needed Accusative is negated, like here, you need Genitive. Genitive doesn't care about the notion of (non-)animate, luckily. So masculine is "tego", feminine is "tej", neuter is "tego".
These were translations of "this", when talking about "that", you just put "tam-" at the beginning.