I don't really have much experience with ancestry, but believe that weighted DNA includes only those matching segments that are passed down from recent common ancestors (6 generations or fewer). Unweighted would also include segments that are common from more ancient ancestors. I think the way they have determined what category a segment is in, is by looking at their length (over generations fragments become smaller) and also at the frequency they saw certain segments in a random test group.
You can read more about it on this site.

How high? How many in centimorgans/ percentage?

With close matches, i.e. siblings and first cousins, you will share a lot of DNA. For example I match 12.2% with my first cousin, or 863cM. Our shared ancestors are our grandparents.All this matching DNA was passed to her and to me from this set of grandparents.
The presence of non-specific short and ancient fragments won't affect the height of my match with her, at least not very much

Kitty Cooper blogs about DNA and I find it very easy to understand
blog.kittycooper.com/2019/07/using-the-new-ancestry-dna-match-features/

This tool is also really useful to putting in the Number of shared Centimorgans so it matches the possible relationship
dnapainter.com/tools/sharedcmv4

That does sound complicated
But great you are not giving up.

I have lots of people who are several relationships appearing all over the place.
For example, a grandparent, aunt/uncle and half sibling will all match (on average) 25%. The DNA testing company will not know which of these options is the correct one, so may either list them all, or make a guess.

Some branches have matches where I know the relationship but they are hundreds cm higher.
DNA is random so you sometimes see odd things. For example, one of my 0.1% matches is a fourth cousin, also matches my cousin but with a whopping 2% (2nd cousin levels). I suspect that she may also have a common ancestor with this match on her mum's side.
So higher than expected matches may arise when you share not one but multiple sets of common ancestors with someone. I came across an interesting one in my own tree where siblings from one family all married siblings from one other family. Their children are double first cousins: instead of sharing one set of grandparents, they share all four grandparents.

I don't know anything about this stuff, as you know @Devlesko, but based on what's been said so far, is the issue that, because of the endogamy of yesteryear in your ancestry, there are a lot of people that you share a fair bit of DNA with, but in many cases the shared DNA consists of a bit of this from here and a bit of that from there, and it adds up to quite a bit of overlap, but because it's all in little bitty bits from multiple sources, it's not very helpful in trying to determine family relationship based on centimorgans?

Or to put it another way, if you had a 'high' match, say around 1000cM (I'm making numbers up here, sorry if it's nonsense), for most people that would mean a close family member, but for you, because the 1000cM is made up of 50+50+100+20+20+50 etc etc, it looks like a high match but is actually multiple low ones. So maybe the weighting system Ancestry uses is a way of telling you which matches are genuinely high ones and which ones just look like it? (Except that like everything else on Ancestry, it may not always be as useful as it appears to be, given that their algorithm probably has imperfections.)

I'm not sure if I'm saying anything you didn't already know. It's a question as much as an answer tbh. But I think because of your Romany heritage, the results may mean different things for you than for non-Romany people. (And given this, maybe 58k matches actually involves fewer than 58k people - ?)

Every day's a school day on this board...

OK I'm not an expert either but I do understand the basics.

This is a handy chart.

fh.familysearch.org/system/files/team/ait/images/blog/centimorgan-chart.png

You share around 50% of your DNA with each parent, and 50% with a full sibling. Each move you make away from that core family unit, you're reducing the percentage of shared DNA. So you might have around 3000 centimorgans in common with a sibling or parent. But as you go through the more distant relations of cousins, second cousins, third cousins it gets more and more remote.

My very best match on Family Tree DNA shares 176 centimorgans with me. He's my 3rd cousin - his great grandfather and my great grandfather were brothers. I have a fourth cousin match and we share 97 centimorgans.

I might have the wrong person but are you the one with Romany ancestry? If you are then the issue is that the genes are not "diluted" with each generation as they are in other communities. So you can't apply the same percentages to Romany communities as you would to other communities. If someone marries their cousin, or even second cousin, then there is less new DNA to share around. Add to that the problems of lack of documentation and you start to really struggle. And yes, you're going to get someone showing up on Ancestry or Family Tree DNA as a much closer relative just because the community gene pool is narrower.

I really do think you need expert help on this. That doesn't mean paying though, there are lots of interested amateurs who love having a puzzle to solve. Have you posted on UK based genealogy websites? There's a UK based facebook group called DNA help for Genealogy (UK) and several other worldwide ones too.

No. I wouldn't call myself an expert or professional either, more of a fanatical but reasonably knowledgeable amateur ...

I know it's a different background but you might check out some of the groups for people chasing down Jewish ancestors - that community has all the same problems with intermarriage and a lot of the disentangling advice will be the same.

good luck!!

I have a match of 822cM and I swear I can't find this person or how they fit in

When you have a match with someone, are you able to contact them? Do you literally not know who they are, or is it that you know who they are but can't see how you're related? I'm wondering whether they could help you figure it out based on what they know about their connections.

Do you tend to find that most people who have done a DNA test have also done some prior family tree research, or do people often do a test as a first step?

That's a question for anyone really...

I'm quite DNA curious but can already see from my tree that the number of potential matches, especially low ones, would be huge just because families were so big in the past. I think I would find it super frustrating to have a match with someone and have no clue how.