Torgamous did a good job on answering, but I'll throw in my $.02 still 
I saw that in the tech tree, the last research (either military or civilian) only requires 8 labs of the respective type, so would it make any difference if I build more than 8? Speed up research or something? Or is it just a waste building extra?
More than 8 is basically just an insurance policy. You don't get any bonuses, but all of the superweapons, ship hulls, and building types that you unlock through research require you to maintain the number of labs they needed to research. So, if you have 8 and build a bunch of superweapons and then you lose a planet and a lab gets destroyed, you can no longer use those superweapons until you get back up to 8 labs. In most cases, though, as you expand your labs will end up deep in your territory so you don't really have to worry about it.
If you're playing a multi-system map with 10 players - just say the one I mentioned in the previous post - what would your fleet be composed of as a TEC? My concern is, I dunno if I should just leave some of my systems alone, without any ships there. If I dun do so, I end up having to split my fleet smaller and smaller to cover all the systems. I only have 3 at the moment, and it is already cutting my defensive fleet quite thin (I allocated 750 points in this fleet, for the rest 1750 points, I made up 2 fleets as attacking force. BTW, I chose large fleet size in the game option).
What I usually do is maximise the static defenses around all the planets linking to the star, and my single defense fleet consists of just long range frigates (whether I'm playing TEC or not) to minimize the fleet points spent and maximize survivability, since they can hide behind all the turrets and among the 3-4 repair bays. But I find that most often the AI attacks the system I'm currently engaged in, once I leave my "home" system and start attacking AIs and capturing planets in another system, it's those new colonies that get all the action and my big fleet just happens to be in the neighborhood Is this also the case for you, or do you get your fully conquered home system frequently attacked while you're fighting in another?
3rd question (I feel myself annoying already, hehe). Do I need a broadcast center in every planet I colonize? I mean, since culture doesn't seem to go to another system, if I solely own a system, I only really need to build one in one of the planets, right?
Personally, I feel that one on each planet is a waste of money. Culture has a range longer than one jump, so I usually spread them out quite a bit. Some of the AIs like to build 4-5 of these around a single planet, which requires me to do the same to beat them back. Basically the way culture works is its rate of spread gets cut every time it passes a planet. If you build one center, it will usually run out of steam after 2-3 jumps and crawl too slow to spread much farther. If you build 4 centers, it will be able to spread past the 2-3 jumps much faster because you're generating 4x as much. So, there's no magic formula for how many to build. Play it by ear. Make sure all of your planets are covered for the economic bonuses, but don't settle on a strict "1 center per x planets". If you see that it's crawling too slowly towards some part of the map, build a few there. Things like that
My last question (finally
) is, how do you do smart ping as said in the manual? I want to tell my allies I'm going to attack this place, have you send in some troops too if avail. Or I want them to come defend my planet. I know it is not really useful as allies may not come, or coming too late, but it would be good if they get there in time. Not because my force is not enough, I just want them to take the casualties for me (I know I am evil, lol). Anyway, I have this smart ping thing working sometimes and most not, especially when I want them to come to my planet. All shown in the action grid are the usual commands like planet development, building civilian things, etc. So, how to get this done correctly?
To ping your allies, select any of their ships, and the three buttons for Attack/Defend/Move will show up in the action grid like you say. They all seem to be focused around planets, so when you give these orders you're basically asking the AI to go to a specific planet. So showing them to attack an enemy fleet around your planet wouldn't work because it will try to "attack" your planet, so you'd need to ask them to defend. Giving a move request before the enemy gets there might work, I don't use it much but I seem to remember that they don't stick around very long after the move. Also note that you can't see the alliances the AI formed, so even though he's allied with you, he might also be allied with the one attacking you 
Hope this helps
