I have no idea about the fake part . Real linen is a fabric that can be tempermental I would do grommet ones before I would add black tabs and if you are worried about how they hang there are weights you can add to the hem to help. IMO the fuller the drapes the better they hang so make sure you get at least enough panels to make them 2 x fullness.

Curtain weights. Also, training the curtains into the shape you want will improve their looks.

If you want them to be inexpensive, use an unlined synthetic or cotton blend, woven to look like linen.

Cheap curtains never hang right. For a curtain to hang properly, it not only needs to be lined, but also interlined. A well-made curtain is worth its weight in gold.

Anglophilia is right. You won't find what you're looking for in cheap readymades. The photos you're talking about, are most likely custom.

Here is is an article that gives some great tips on hanging linen drapes. I think the editor of Atticmag posts occasionally and Allison who posts here frequently used to be affiliated with this on-line mag. I don’t have pure linen, but a linen blend from Pottery Barn in my LR and the extra clip rings do make a significant difference.

https://www.atticmag.com/2018/05/ready-made-curtain-hack/

I work in a drapery workroom and we have altered panels from Pottery Barn, Restoration H'ware and others. They are never as well made as custom. That being said, if custom is out of your budget, I would consider a nice synthetic over 100% linen. Will you be opening/closing, or do you want stationary panels. Back tabs are fine if you're not going to open and close. Yes, they should be lined.

Linen degrades rather quickly when exposed to sunlight. While I love linen, I would not use it for drapes, unless they have a black-out lining. I line all the drapes that I make, and I often line curtains as well, depending on the fabric.

A linen blend will hold up better than 100% linen. Rayon is not a particularly good choice either, for durability. Normally I'm not fond of synthetic fabrics, but I think they are good choices for drapery and carpet, and sometimes a blend is good for upholstery. That said, I do NOT like working with synthetics.

BTW, I had a birthday lunch with Jane from atticmag a few years ago when she was in L.A. visiting relatives. She's quite delightful.

Thank you everyone for the helpful insights.

I think I am going to take a look around at some synthetics in store and compare them against what I found in linen.

Also- I found a relatively inexpensive custom place on Etsy with good reviews. Could I trust custom linen drapes that start at 58$/panel?

Make sure that's not $58/width, meaning if you want a fuller panel that takes 2 widths of fabric, the price may be $118/panel. $58/width sounds more correct (could be more or less depending), but $58 for any size panel, I'd be a bit more cautious, but some people just charge less.

Shadylady- the window is only about 41 inches wide, so I think I’d only need one standard with in any case (right???)

Much depends on your budget, and also the look you're going for. If this is for the room with the daybed, where this is your inspiration photo,

I'm not sure that lined (esp heavily or blackout) curtains will go with the airy, coastal, British colonial look you're aiming for.

I have several windows with Ikea's Aina curtains, which are 100 percent linen and find they fall/hang very nicely. I would opt for linen, and rings with drapery hooks. Rings with hooks, vs. back tabs or even rings with clips, is a huge start toward having curtains hang properly.

One width panels will be fine for that size window. If you're looking for unlined, airy panels, just ask them if they can put in the chain weights that someone mentioned. It will add to the cost but will help it hang better.