Neterw,
Ok, lets see if i understand correctly.
Non-believer?
haha, thats ok, i dont mind, we will try to work through this.
Change the mesh after activate the new ability for mini-mesh? Actually there is NO ability that changes meshes, i know the starbase does that in Entrenchment, but im not aware of it being possible in Vanilla, and that is NOT what i am doing here. I changed the mesh period. It is invisible, it stays invisible, it doesnt change back and forth.
No matter how mini a logo is there is a big circle around it? In default mode, icons for ships are ON, if you go to MENU and then to EFFECTS you will see the option for ICONS and OWNERSHIP ICONS, if this is unchecked, then the icons in the game are removed, UNLESS the ship is selected, then it shows. (Nice for finding out where your ships are).
Global warning tiger for jumping to enemy territory? How to bypass it? Without making disadvantages to non-cloaked fleet warnings? There is a warning for arriving too?
I dont understand the tiger part, but Yes, jumping to enemy territory will give a warning, that is part of the game. How to bypass it? Cant bypass it that i know of, well.. that is not true, you could probably remove it from the string dialog, but why? We dont want a fleet that cant see us, or attack us, only a "Visually Pleasing" difference in the gameplay. Yes there is a warning on arriving too. Also part of the game. I see no reasonable reason to take it out. It is there for a purpose and certainly all the little visual and audio clues i can get for "Where" my invisible fleet is the better. That is of course this modders opinion.
Auto-attack is working! No matter how small the mesh the ai will attack the cloaked ship.
Yes,
That is standard behaviour. There is an ability that i gave the first invisible ship, to phase out, but i modified it to not cut out engines or to disable weapons, and the modified version affects friendlies as a protection mode, when phased out the invisible ships become "UNSEEN" by the enemy. This was of course made by the Devs, a very nice ability, but i didnt want it to make the Inviso ships too powerful, so i set it to a two minute recharge. So its there, and protects them somewhat, but not completely and not continuously.
And i dont really want the enemy to NOT attack my fleet. That of course is the purpose of the game, to engage the enemy in battle and to use strategy and tactics to win, or lose. But it is a game, for fun. I DONT want the enemy to NOT attack.
These really are NOT issues i think should be corrected. What you are asking for is an invulnerable, unseeable, unattackable fleet, which in my opinion, doesnt sound like very much fun.
You are welcome to CHANGE my mod, this community mod, because i wanted it in all the people's names that were on the Cloak thread. And you can change some of the behaviour you are talking about in the string file in the string folder, or by removing the icon calls in the entity files, or by using abilities to make the enemy not attack them. All this should be very DOABLE. 
But that is a project i will leave to you, any others that wish to change the mod may do so as well, just credit the Cloak Crew for the base you are taking from, and change it to your hearts content. 
Sorry to disagree with your premises, but i just dont see it the same as you, but you are most welcome to change it, and it looks workable. Have fun and keep modding!!!

-Teal