Avoiding/getting rid of topological errors after dissolve?

You could try using the GRASS function v.clean and selecting the rmdangle which aims to remove any annoying dangles. You could access this from the Processing Toolbox.

You can set the "threshold" option which determines what dangles can be deleted:

The rmdangle tool deletes a dangle if the (combined) length is shorter than thresh or thresh < 0. If the combined length is larger than thresh, nothing is deleted.

Instead of remove dangles you can correct the original layer. Use v.clean like Joseph says but run the tool v.clean.snap on your original borders layer, set the tollerance (for borders I think you can use a big value but you have to try) and then try again to dissolve.

You can find eventualy remaining dangles using Topology Checker and "Must not have gaps".

For me this worked: processing toolbox > 'Delete Holes' and run without any minimum area.