Last year during the summer, I would go home on weekends.
As OP correctly says, past habitual - he usually/always went home on weekends. No problem.
Last year during the summer, I would have gone home on weekends.
As TimLymington says, native speakers might be tempted to interpret this as an incomplete statement, with the unstated continuation "...but because of [some excuse] I didn't". But that's only because we're struggling to find any way to make sense of a rather unlikely utterance, so I suggest we consider a simpler one...
Last Friday night I would have gone to the pub.
Again, it's possible there's an unstated "...but my wife wouldn't let me."
But if the speaker had just been asked "Where were you last Friday night?", it would be a perfectly normal and complete reply. In such a context, the speaker may be implying that he's normally in the pub on Friday nights, so he would have been there on that particular occasion. Or he may be implying that although he's not absolutely certain, it would turn out that he was there, if the matter were to be investigated more thoroughly.
Last year during the summer, I would have completed three projects every month.
This one somewhat complicates the issue, because three projects every month automatically creates a past habitual context. Native speakers don't feel tempted to assume there's an unspoken "...if I hadn't lazed around the pool every day.", though they might expect a follow-on along the lines of "but I think I could only manage two projects a month this year."
The fact of the matter is that whatever the textbooks say, people often use would have [verbed] in that way. Semantically, there's no real difference between OP's version and...
Last year during the summer, I would complete three projects every month.
Last year during the summer, I completed three projects every month.
As pointed out, these are not good examples for getting to grips with usage of would [have], because phrases like during the summer and three projects every month blur the focus on exactly what time we're talking about, and what exactly might have been habitually repeated.