LIghtroom can store edits 1) Directly inside the catalog 2) In sidecar XMP files for RAW files 3) In XMP format inside supported files (DNG, JPEG, TIFF and PSD) 4) In XMP format inside exported files (JPEG, TIFF and PSD).
I'm confused on this issue...I always check the Catalog Setting: "Include Develop settings in metadata inside JPEG, TIFF, PNG, and PSD files" But it isn't happening apparently. When I open a TIFF outside of Lr that had a brush applied, the brush edit does not show. Same lack of application on...
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Usually only Lightroom and Photoshop (through ACR) can read and display those edits. Some third party tool could read them as well, but as long as data are for Adobe proprietary algorithms. there's no warranty they are processed exactly the same way. Tools unaware of the sidecar/embedded data will display only the original image without edits.
If edits are stored in the catalog only, you need the catalog as well. For sidecar files, you need the .XMP file also. For files with embedded data, you need only the image file. Anyway, when moving among LR systems it's better to move the catalog anyway because LR is built on it and can't work without one, and you don't have the catalog, you need to re-import images into another catalog - and you would lose everything else stored in the catalog (collections, virtual copies, additional metadata, tags, etc.)
Yet because catalogs may be system-specific (i.e. file paths), you may need to export/import catalogs to ensure everything is correct, and that can be time consuming for large catalogs (you can have multiple catalogs, and export/import subsets). LR is not really a multiuser application. But if two systems can access the images with the same path (i.e. external storage), then the same catalog could work. Beware anyway different catalogs aren't kept in sync automatically.
When you convert to DNG within Lightroom, the original RAW won't be deleted, but only the DNG will appear in the catalog as the "original" file.
The Import pane of Lightroom will let you choose the file destination, and how to create required folders. Once they are in the catalog, it you move them from inside Lightroom it will update the catalog data automatically, and move sidecar files also. If you do it outside LR, you will need to update the catalog manually from Lightroom - i.e. it will tell it can't find the files and ask you for the new position. Same for deletions.
To work with RAW files with PS they need to be opened first in ACR (which shares the processing engine with LR) - PS will do it automatically when you select a RAW file, and then imported into Photoshop (as a bitmap image or a smart object) - then you have to save your edits to another file. As working for Photoshop means you're probably going to use layers, smart objects etc. to take advantage of its full power usually it means you want to save those also (but for final output images), so you need to save in a file format fully supporting that, usually PS its own PSD format. For final outputs you usually want to "flatten" (merge all layers into a single one, and all edits), resize, and then save to JPEG, TIFF, PDF, etc. as required.
PS is designed with a broad range of image/graphic creation and editing needs in mind, making it more complex to learn, while LR is designed for photographers only. It's quite easy to work in LR, export to PS for any editing needs that can't be done in LR, and then back for image management. Some people work that way, others only in LR, others only in PS. You have to find what works best for you.
PS can do a lot more than LR and with extreme control, yet you can edit one file at a time (you can record edits and apply them in batch). LR allows for "synced edits" where you can apply the same edits to a set of images at the same time. LR allows to create easily "virtual copies" also, and apply different edits to them without duplicating the original image.
Some PS edits are "destructive" - after the image is saved they can't be easily rolled back. LR edits are always non destructive, but you need LR/ACR to see them, or to export to a new image with all edits applied permanently.