So we basically need to update to 0.95+ both in the online and offline computers if we want to avoid compatibility issues, right? Could you please ELI5 the kind of issues we could face if we do not update?
You won't need to update the offline if you are not planning on using segwit.
If you do not update to 0.95+ but are using 0.13.1+ after segwit deploys, armory will simply not work. It will not be able to parse the blocks that 0.13.1 writes to the disk. IIRC it will simply hang at the initialization and look like it is doing something when nothing is really happening.
Furthermore:
Are there offline bundles for 0.95+? I can only find offline bundles for 0.93.3.
Thanks.
No. There won't be any offline bundles until the release containing segwit (which should be 0.96).