Writer automatically draws random black boxes after some words

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Thanks for providing the additional detail. It may be a limitation in the encoding in the Liberation Serif for the Unicode range containing Georgian. The empty rectangles do look like a placeholder character for an unencoded code point. Examining the copy of Liberation Serif (v2.00.1) provided with v5.1.1.1 indicates these missing code points:

• U+10c7 through U+10cf
• U+10fd through U+10ff

There is a mismatch between these uncoded characters and that specified in the Unicode v8.0 chart for Georgian which has blanks in:

• U+10c6
• U+10c8 through U+10cc
• U+10ce through U+10cf

Perhaps the problem relates to this mismatch? The fact that you cannot select the rectangles is strange though as generally even an unencoded character is still selectable. This is why I though it may be an automatically inserted grammatical addition, such as a space or word joiner (which is still a possibility, given it appears to only be occurring in LO).

The "black boxes" look like placeholders as used for U-codepoints the specific glyph for is missing.
In fact they do not represent any character or codepoint, not even a not visible one. They are random artefacts caused by some "accident" during rendering next to a break in textflow.
The attached example is containing three specimens, two of them preceding a hard linebreak, the third and last one preceding the paragraph end. In all three of these examples the "black box" vanishes as soon as I insert an ordinary space between the last shown (Georgian) letter and the non-printable break. It also often disappears if I insert an additional space, some other whitespace or even a zero width space (U+200B) elsewhere in the line concerned.
Thus I would judge we view consequences of a strange error (bug) related to calculations concerning the metrics of lines during rendering. On my system the bad display is also viewed in print preview, but does not actually print. (LibO V5.1.0, Win 8.1, Kyocera FS-1030D with KX driver)
(Tried also with Apache OO 4.1.2. It spoilt the text completely. Rendering Georgian writing seems to be a difficult task. As I do not know anything about the implementation of metrics in fonts, I cannot help any further. Either the OP or someone knowing a bit more should file a bug report concerning the issue.

Editing: I also copied the Georgian text posted in the self-answer of @rebar into a virgin 'Writer' document, eliminated the textflow marks and replaced them with 'Writer' standard. This way I also got the bad view. It seems to be fully reproducible.