For starter, when I take over another empire's colony in a war, it keeps influence of the original empire rather than changes it to mine. Makes no sence to me. At the very least the colony should start emanating my zone of influence, gradually replacing the original one, IMO.

What is worse, since the colony remains in the other empire's zone of influence, I cannot defend it from an invasion after a truce is forced on me.

Another strange artifact of this mechanics. If you fight a war for a colony located in a third empire's territory, the rules of engagement are determined by you relations with that third empire. Which may make some sence, but it's quite frustrating not to be able to protect your colony from an invasion because of it.

One thing is strange about influence is the absence of ways to spend it, when you have them in abundance. The costs of senate laws don't scale up after all. Adding a technology to spend influence points, for instance, to boost you growth by attracting immigration from neighboring empires would be nice.

Another one about Vodyani. Their relations with minor factions are puzzling me. First you spend loads of influence points early in the game to integrated a faction (and complete an empire storyline quest). They you have to spend 100 more to decolonize the system? Isn't it a bit doo costly, compared to simply ousting them?

BTW, why does declaring a war on a minor faction require a technology? I see why from the investment\profit standpoint, but narratively it makes no sence.

Other impressions.

Love the replacement of a tech tree with EL-style eras. More choices is always good. And since it is tied so much with colonization, the importance of choosing the right techs is even bigger.

Combat is fine, assuming that more engagement options would be introduced later. But it would be really nice if you could get a battle log as an alternative to watching a cinematic battle to evaluate the fleet performance.

Exploration and probes work really well, better than I expected. It would make sence to add exploration points for systems with no planets. And I hope for more exploration outcomes in the future.

System improvement is OKish. I don't see why you shouldn't be able to have more than one option for improvement on each tier so that you could better customize your empire. Probably, as a technology.

Other suggestions.

There should be a way to distinguish planets according to their habitability tier in the interface. Add a corresponding number of dots orbiting the circle, probably? It helps chosing systems to colonize -- and picking technologies for it.

The introduction of a newly discovered system should have an option to turn off. After a 100th time it feels waste of time, not an eye candy.

More options to spend unused industry in developed colonies. Use it to speed up creation of new trading companies. Export a portion of it into neighboring colonies. Boost manpower generation, etc. The same goes about planet specialization.

I agree entirely with your observations about influence. Currently the border system does not work. And at higher difficulties this problem is escalated even more. Essentially it forces you to invest in influence tech since without it, taking over enemy systems is pointless. This is because enemy influence is only removed by your own... from what I can tell anyways. 

Once you remove the source, it should also remove all the gained influence. But I expect this will be adjusted in due time, since it simply does not make any sense. 

For colonies you take over the "influence circle" should chance colour to indicate this empire no longer has effective influence there (so that your ships can travel).
And the circle should shrink (there could be different options here).

1) Quick original influence reduction but rebellion on planet and population loss.

2) Slow reduction and maybe an approval bonus for being a nice new master.

Apart from the original influence getting smaller your own empires influence would start to rise there (based on influence generation).
Once they even out the influence zone it becomes your zone.

Until that happens it is a conflict zone for both you and the other empire.

When a planer is totally destroyed (all population killed) the influence zone should be gone very fast.

I wouldn't remove it right away so that recolonizing the planet quickly would make sense but 8 turns is the most.

I also notice that the unique planets generate influence in a huge amount and this should be reduced.

And the taking over of one influence zone by another looks to quick.

I understand the "peacefull take over" needs to happen somehow but I would make reducing an existing influence zone much harder.

It is just like with boarders. They don't change so easily.

I honestly do not mind that unique planets have special bonuses of that kind.. they are after all filled with endless stuff. So it makes sense. However they should not be easy to colonize.... I hope to see them as vital strategic points on the map that you will want to fight over. 

However I do think that they should be like era 3-4 kinda deals. Assuming they stick with an era system. 

But overall as long as influence loss is swifter, then I do not really mind that gaining it is fast as well.