In a nutshell:
1) I don't memorize them. I couldn't possibly, although I'm sure there are some very seasoned astrologers who know the tables by heart. I unashamedly use a cheat sheet. Whatever works, I say.
2) It depends on the chart. Most charts can be interpreted within a few minutes. For complicated ones, maybe fifteen minutes. The symbolism of horary makes it easy to pick out what 'needs' to be read and what can be ignored. And honestly, for most charts you don't need to worry about more technical things like contra-parallel and antiscia. IMO if you have to fuss with this sort of stuff in a chart, there's something wrong with the question, like, the querent doesn't really know what they're asking, or the astrologer is trying too hard to complicate things
. Beseigement is easy: it's just a significator trapped between two 'unfriendly' planets where it can't act, usually but not limited to the malefics.
3) The Parts are a matter of preference. Most horarists will use the Part of Fortune without a second thought, as it indicates what the querent 'values' (often a very interesting insight into the context of the chart/question). As to the other Parts, I personally leave them out unless they are specific
to that question, so, Part of Death in a question about a terminal illness, etc.
4) My bit of 'wisdom' for you is: practice and keep it simple.
Practice on charts where you already know the answer (the Skyscript site has a whole tutorial on this, which is excellent). Practice on the charts we delineate on here and see if you can see what the other students are looking at, and why.
Keep it simple, especially when starting out, by omitting everything in a chart that doesn't pertain to that question, especially (IMO) asteroids and the outer planets. They are nice embellishments, but not necessary to get a clear and simple answer to a horary question. When you've got the 'inner planets' (e.g. Sun, Moon, Mercury, Venus, Mars, Jupiter, Saturn) nailed, and understand their natures and what their capabilities are in a chart, THEN you can start including more fiddly bits to see whether they add anything.
As I've told other students, in horary, it's not the aspects that count, it's what the aspects are DOING that matter the most. Are they applying? Separating? Going void of course? Interfered with by another planet? These things are the meat and bones of a good horary interpretation.