In the first case, you wouldn't lose any rights to your code. What you'd lose would be the right to copy and distribute the LGPL'd code, or more precisely you wouldn't be granted that right in the first place since you didn't comply with the license and it's grant is conditional on that. That would then allow the author of that code to sue you for copyright infringement for copying and distributing their code without a license to do so. The easiest way to avoid any problems here is to either distribute the original dependency package or to simply note in your documentation that the dependency is licensed under the LGPL and provide a link to wherever you got it from. If it's a standard package on the platform, eg. any of the LGPL'd software in Debian Linux, you needn't even distribute it, just note it as a prerequisite and require the user to install the packages before installing your program (most packaging systems let you automate that by declaring the dependency and letting the packaging system automatically do the installation of any required packages).
In the second case, if they just change the code it won't make any difference. It's only if they start distributing their modified version that they'd run into the same issue of copyright infringement that you'd run into in the first case. This shouldn't involve your code at all, you said the dependency was LGPL so it's a separate shared-object library or something similar and (aside from the small bits of declarations and constants) the code's entirely in it's own files not physically part of your program.