I have a few questions about hitsounding beatmaps.

1: I have a skin which I really like the hitsounds for. How would I set these as the custom hitsounds for my map and have them be the hitsounds everyone else hears when playing the map

2: How do I download custom hitsounds. I have tried downloading hitsounds from the hitsound library, but the instructions were very confusing and I couldn't figure out how to apply them to the map.

Hey there.
Good job at articulating clear questions that can be answered easily!

Before answering your questions in specific, let's talk a bit how hitsounds work in osu!:
osu! uses 3 different sets of hitsounds that are referred to as "soft", "normal" and "drum".
Each of these sets consists of 4 hitsounds, the so-called hitnormal that is played on every note that has no or only 1 additional hitsound and the additional hitsounds "whistle", "finish" and "clap".
When skinning or working with custom hitsounds you need to supply hitsounds relating to these terms.
The pattern for naming these is the following:
<set>-hit<sound>.wav
So for example normal-hitnormal.wav or drum-hitclap.wav

1. In order to use hitsounds from a skin for a beatmap you need to extract the related audio files from the skin first.
To do that, enter the Options and scroll down to click "Open current skin folder" while you have the desired skin selected. In the skin folder you should be able to find hitsound files according to the naming pattern mentioned above.
You can copy these and insert them into the folder of your beatmap and osu! will recognise them and use them in place of the default hitsounds.
Note that it is considered poor behaviour to upload hitsounds from a skin without asking the creator for permission to use them so if possible seek them out before you upload your beatmap with the hitsounds in them.

2. I'm not sure what instructions exactly you tried to follow, was it this one?
Essentially osu! recognises the hitsounds in the beatmap folder if the format is correct. That means if you have a hitsound which you want to add, you simply rename it to the hitsound it should replace.
If you need to add more hitsounds you can work with different hitsound sets by adding numbers to the filenames such as drum-hitnormal2.wav and setting the sampleset in the timing tab as shown in that guide.

Feel free to ask further if something is unclear or you need more information :-)

Thank you so much! Your explanation was very clear and much better than all of the hitsounding videos I have watched. I noticed you helped me on something else and you are also very active on the forum. I just want to thank you for putting all this effort into helping people!

One last thing, the creator of the skin I am trying to use hasn't been active for a year. If they don't respond to my request, would I be better off finding different hitsounds to use?

You're very welcome, helping is a happy thing with people like you who think about their problem before asking a question and also get back for a thank you :-)

Generally speaking, clearance of copyrighted audio or image material is always a somewhat problematic thing if you intend to upload a map and likely one of the painful spots of the osu! mapping community. I'd recommend you to familiarise yourself a bit with the Beatmap submission rules and the osu! Copyright Policy and weigh up the implications and potential consequences of uploading copyrighted material to osu!.
It is very difficult to judge what percentage of beatmaps actually has obtained permission for all their resources and you will need to figure out yourself what you consider acceptable for uploading.
However, if you only want to use material for private use without publicly sharing the map you don't need to worry about this at all.


Anyway, on to the question:

Based on what I wrote earlier, not using them is obviously safer...however if the creator does not respond to your request you can still try to search around a bit by asking yourself questions such as:

Where has this skin originally been uploaded?
Potentially the creator left comments on usage permission and/or another source they took the hitsounds from. The place of the original upload might also give other cues:
If the skin has an official thread in the Completed Skins subforum it might be fair to assume that the skin creator intended the sounds to be used for osu! and use them anyway.
Some skins - although rare - may also include a textfile in the skin folder that specifies the sources of elements not originally created by the skin creator.

Are there any clues in the audio-metadata of the hitsoundfiles?
Sometimes they contain the creator of the file which might in some cases even be osu! users you can then ask instead (Example).

Whether you end up using the hitsounds or not, making an effort to actually find out about the copyright situation of given files will give you a better feeling about your upload. How far your efforts will go and how you act upon the results is something you need to decide for yourself.