Yeah, it's not easy exporting artwork from CorelDRAW over to Adobe Illustrator. Generally speaking, artwork that is more complex in nature will have more problems when imported into Illustrator.
I really wonder if the software developers at Corel (aka "Alludo") are doing any quality control tests on how their PDF, EPS and AI export filters perform when the resulting files are imported into Adobe Illustrator and other rival graphics applications. I've used both CorelDRAW and Adobe Illustrator for many years. Over the past decade it seems like the list of technical hang-ups have grown.
To be fair, Adobe Illustrator can't import or export CDR files. CorelDRAW can run into plenty of problems trying to read AI files. That's one of the reasons why I've had to use both applications for all these years. CorelDRAW is still very popular in the sign industry and some other niches. I receive CDR files from certain types of users. But most corporate branding work is done using Illustrator. The large format printing RIP applications we use are Adobe certified and tend to produce better results when fed Illustrator-based PDF files. There are lots of features and live effects in Illustrator that have no compatible counterpart in CorelDRAW. The same is true going the other direction. The rival applications do not overlap each other completely in terms of features.
IMHO, the Illustrator AI CS6 filter in CorelDRAW is the best choice to use for exporting artwork to Illustrator. The export filter has its technical problems, but issues get worse trying to place/open Corel-generated PDF or EPS files in Illustrator. Corel's PDF export filter works well at generating files for print-only use. The PDF files are not edit-friendly (unlike Illustrator PDFs with AI editing preserved). EPS has its own issues, plus EPS does not support transparency effects.
While Corel's AI export filter might be the best choice, a lot of repair and re-building work will be involved unfortunately.
Live effects such as drop-shadows should be re-created in Illustrator. Look out for gradient fills on objects getting knocked out of kilter, especially if the AI file was exported from an older version of CorelDRAW. When CorelDRAW exports objects with transparency effects it sometimes turns the fill into a rasterized, pixel-based object. This is definitely true for gradient fills that have varying levels of transparency.
The issue with text is pretty maddening. Single lines of "artistic text" usually import into Illustrator with no problem. Line spacing always gets messed up on any imported multi-line text objects. And you can't simply repair the text by changing line spacing values in the text properties palette. To fix the problem I'll create a "dummy" piece of area text or point text in a new, clean Illustrator file, style it how it should be, then copy the text object over into the file with the imported artwork from CorelDRAW. I'll use the eyedropper tool to copy the properties of the correct text block onto the errant blocks. Usually that repairs the line spacing to a non-corrupted value. The approach can be a real pain in the ***. If there is a lot of text, I'll just copy the verbiage from the CorelDRAW file into a notepad document and then copy that plain text from Notepad into Illustrator. Then I just re-do the styling and placement in the layout.
Astute Graphics' Vector First-Aid plugin has a feature that can repair strings of point text and area text that have been broken apart into lots of little pieces. Astute Graphics' plugins are great, but they're not cheap.
Oh, one more thing, I usually do not run CorelDRAW and Adobe Illustrator at the same time. There is a technical problem with fonts. If I have CorelDRAW running in the background while using Illustrator I'll see some styles in some type families disappear from the font menu. These are fonts I have installed directly into the Windows Fonts folder, not using Corel's Font Manager. It's weird. If I close both applications and then re-launch Illustrator all of my fonts are restored.
There are plenty of things I do like about CorelDRAW. It's a breeze to use for creating technical drawings. But if the artwork in a layout is going to involve a lot of complex fills, transparency effects, etc I may choose to create the layout and shop drawings using only Adobe Illustrator.