There is actually no problem, everything is working as expected.
The automatic prompt that MiKTeX throws up when a missing package is detected silently downloads and installs packages without producing much diagnostic information. Most packages download and install in a matter of seconds, and this works fine.
cbgreeks, which is necessary for the functioning of textgreek, is almost 70 MB large. Even if you have a fast connection, your nearest CTAN repository server may not be able to provide a very fast upload rate for you. Therefore, downloading this package takes a long time. And the TeX compiler must wait until the download finishes.
In theory you could just sit there and wait for an indeterminate amount of time, and eventually the download will finish and your compiled PDF will probably appear. For such a large package, however, it's better to use a more informative interface: Go to Start Menu -> MiKTeX -> Maintenance -> Package manager. In the Name field in the top right, type the name of your package (eg. cbgreek). The list should be filtered to show the package you want. Right click the package and select Install.
You will now be able to watch the progress of the download and installation. If the problem is indeed the package size, you will also be able to observe more clearly just how long the package takes to install.
After installation of your large package(s) is complete, TeXstudio should be able to compile normally.
the package name underlined and a pop-up says it's not found
With a MikTeX installation, TXS checks the presence of a package by analyzing the result of mpm --list. I assume that this does not cover packages in a local tex folder. You may disable package checking at Options -> Editor -> Inline Checking -> Packages.
There are some other files in there that look promising like "miktexPackageNames.dat."
This file maps the MikTeX package names to the .sty and .cls file names. It's used as cache for a fast lookup. It's automatically updated in a background process, therefore, manual modifications do not persist.
It is a little surprising that C:\Users...\AppData\Roaming\texstudio\IEEEtrantools.cwl is there.
TXS tries to detect the commands defined in a style file. These are stored in a .cwl file. See the TeXstudio FAQ and the link therein for more details. Depending on the complexity of the sty files, it is not possible to extract all valid commands. You may add missing commands to the cwl file. To see the changes, restart TXS afterwards.
Best Answer
The Linux version of MiKTeX was always declared as provisional, and is meanwhile not anymore available for download from MiKTeX.org. Therefore you are forced anyhow to use TeX Live, but apparently you used the pre-installed version of Ubuntu. For this issue we have a good Q & A here on TeX.SX: How to install "vanilla" TeXLive on Debian or Ubuntu? (and judging from comments this seems indeed to have helped you).
But apart from this there is still a number of packages, where MiKTeX is more liberal in estimation, what is called free. For several fonts of this type you can execute the script
getnonfreefonts, that is included in TeX Live. For other packages, maybe also some fonts, there is no other solution then to manually install them. You can read for this these questions and answers: Where do I place my own .sty or .cls files, to make them available to all my .tex files? and especially for fonts Manual font installation.Regarding TeXstudio: This is a multi-platform program, there is a version for Linux as well as one for Windows.