Q: What is meant by 'Editor Assigned'?

Your understanding is bang on, well, mostly. ‘Editor Assigned’ means that an Associate Editor (AE) has been assigned to your manuscript. The AE will now go through your manuscript to know what it’s about and whether the study is interesting, novel, and relevant enough to be published with the journal and therefore to be sent for peer review.

It’s been exactly a month since you submitted your manuscript, and so far, it’s reached the ‘Editor Assigned’ status. This isn’t too long, given that this is a Springer journal. So, you should wait for some time. It might stay at this status for a week or so more, after which you’ll know whether it’s going for a peer review or will be desk-rejected. Which therefore means, to answer your second question, that this status does not yet mean a desk rejection. :-)

So, you should wait for some time. If the status doesn’t change in another week or so, you could write to the journal requesting an update. However, given again that this is a Springer journal, their workflow is on point, and you needn’t worry about regular/timely updates.

Until then, it might help to equip yourself with more knowledge about journal decision-making, statuses, and acceptance through the following resources:

• Behind the scenes: A journal editor's insight into what happens post manuscript submission
• Editorial decision-making: what are the possible outcomes for a manuscript?
• Keep calm and wait: A guide to understanding journal statuses [Handbook]

All the best for the next update!