2. When it is stated that the Market Center planetary improvement boosts the "economic output" of the planet by 10%, what do they mean with "economic output"? Exactly what is this economic output?
It is the tax-reward from your planet's population. The more people on your planets, the more increases your reward. As well as the tax-slider jumps here in, and all economic structures built on this planet. The underlying tax-formula to this is horrible, and all you need to know is that with Stocks/Banks built you'll get more money from your people.
Optimally you may want to run your "production slider" at 100% throughout the whole game. However, if you've built too many labs/facs then your spending might consume all your economic input, and it is time to build some banks. So generally, only your civ-wide economic balance is all that matters - how it is calculated exactly on a per-planet-basis is a well-guarded SD-secret
How can it be that in the 1st picture I have 61 flasks (at the top of the Earth colony management screen where production can be focused with the three radio button), and in the second picture, when I go to planet "Summary", I have only 47 flask (I assume this 47 tp flasks mean it is a capacity, and not the actual output).
I am sorry, I don't know how to insert a picture:(
No need for a pic as long as you describe the screen-title from the game as exact as you do
What you see in the "Summary" pop-up is only the planets "base"-production, in your case it is the "research production" (RP) which labs generate, your CivCap and also the Initial Colonies. This base production gets enhanced/multiplied by many many factors (see here) but at your current state of playing I wouldn't try to immerse myself deep into the numbers - you will probably be lost, esp. because there's also a lot of diversity from DL to DA to TA.
To find out whether I understand this correctly, let’s see the following example:
The research cost of the “Xeno Communications” technology is 15 tp. I’ve got two labs, with a nominal capacity or output of 6 tp each. That is 2x6=12 tp capacity in total. If I put both the production spending slider and the research spending slider at 100%, then all the capacity of my two labs will be utilized to max. extent, the actual research output will be 12 tp. That means I can research 12 tp worth of technology in a one week turn. As the cost of the technology to be researched is higher (15 tp) than the output of my labs (12 tp), I need another week to fully research the tech in question. One tp costs 1 bc, so in the first week the research costs me 12 bc. In the second week however, I need only 3 tp to finish the research (that would cost 3 bc), so I need to pull back on the sliders until I get the required 3 bc research spending. If I don’t pull back on the sliders, than in the second week of research the labs will continue to operate at full capacity, producing another 12 tp output that would cost another 12 bc. But I need only 3 tp in the second week, so the remaining 9 tp and its cost of 9 bc would be wasted.
Am I right? This is how it works?
Yes, it works like this, but note that only your base-production costs 1RP=1bc other RP's that you have from multiplication (racial, TechCapital whatever) only costs 50%. And no, the RP's are not wasted - they are taken to research the next technology in that branch on the tree. If the tree is finished, that is, if you just researched an end-tier tech, then your RP's might carry-over to the next branch - or NOT. This is different from version to version, and from branch to branch. Generally there is to say that RP's are NOT lost.
Speaking of this, SP (social Production) is also not lost - unused SP is funded back at the end-turn.
What is however lost is the Military Production (MP) once a ship in your starport is finished. If you build an empty Cargo Hull (55 MP) and your planet has an output of 25MP, then you'll need 3 turns = 75MP's where 20MP's are lost - and you even have to pay for them - they don't get refunded. Hence optimally, you may want to built up your colonies to match your ship-designs, or vice versa, you design different usefull ships that have a different cost, and queue only the fitting into the starport, so that as less as MP gets overproduced.