Is EUPL alone compatible with proprietary softwares?
Short answer: Yes, you can use an EUPL library in your proprietary application.
There are lots of information about this topic on the official website of the European Union:
It all says: virality does not exist legally in Europe, even for the GPLv3 or AGPLv3. And you are bound to the EU law for the code corresponding to EUPL.
Here are some slides from Patrice-Emmanuel Schmitz, a lawyer that participated in drafting EUPL, where he clearly question the legality of "strong" copyleft licenses in Europe.
(To be fair, there is a debate around this, as the lawyers of FSF Europe does not really agree with that. Can't find the link about this anymore though. You can find cases where the GNU GPL seems to have been successfully enforced in Germany.)
While the FSF wishes there is virality and tries to enforce it as much as they can with their own GNU licenses, the EU did not intend that in any way. The EUPL is closer to the weak copyleft MPL or EPL in intention than to the GPL.
And when it comes to licensing, the only person that may sue you is the copyright owners that initially released their software under the said license. Therefore, if they say it is not viral, especially on the official European Union website you can feel safe and use it.
There's a big misunderstanding with the EUPL, probably because of Google treatment of the license (they literally banned it without much explanations).
LGPL library that depends on an EUPL library: which license is applied to the combined work?
EUPL is a compatible copyleft license. What is "compatible about"? It means that you can combine a EUPL licensed software with a LGPL/GPL/AGPL/EPL/MPL software and the licensed of the third-party software will apply.
Example in your case: the library you want to depend on is licensed under LGPL and rely on an EUPL software. Therefore, EUPL somehow disappears and only LGPL applies. Don't get me wrong the initial library is still under EUPL, but as a whole, the two combined libraries can just be licensed under LGPL. It is very closed to dual licensing for the EUPL Library. As a consequence you can treat the library you depend on as a standard LGPL library. You don't need to care about its EUPL dependency.
Some links that may help:
So in your case, you only need to know whether you can link the LGPL library with your application. With LGPL, it comes down to dynamic vs static linking.
Side note on Free software (as in Freedom) vs open-source
I noticed you seem to confuse "commercial", "proprietary", "open-source", "free software" and "closed-source".
This article is a good read to get your head straight about all these terms and know what you can and cannot do.
Disclaimer: I am biased, I wrote it myself.
In short, LGPL is incompatible with proprietary softwares to some extent. LGPL is not incompatible with commercial products. LGPL softwares can be sold for a price. Proprietary softwares are not necessarily closed-source: they can be made "source-available". A free software can be closed-source, as "not available online for the general public". A free software source code is always available to the people to which it has been distributed to, that's all.
Open-Source refers to any software that is distributed under a license approved by the Open Source Initiative. Closed-source is strictly defined as opposed to Open-Source, but can also mean: "not available online for the general public".