as for the number of transactions: when i look at blokchain.info they have nice graphs about the average number of transactions per block and it's about 650 atm.
Is this the average amount of transactions happpening in about 10 minutes and the pools mining a block just include all the transactions during that time in a block?
The way blocks are mined is that information about transactions has to be included (via several steps) into a block header. It's that header that then gets hashed as part of the mining process. So typically the transactions are included only once*1, and then attempted to be mined. So this is just the number of transactions in a block, and then averaged over several blocks. Not sure what window blockchain.info uses though
Is there any incentive for pools to include that amount of txns in a block?
For any miner, the main incentive to include transactions is the fees. A secondary incentive might be 'for the good of the network', especially considering that the reward right now is much, much higher than most fees anyway. A miner may choose to include as many transactions as possible into a block to collect the highest amount of fees*2.
i see that it it useful to include as much fee-txns as possble but what about the ones without a fee, why not just pick the ones with a fee? i assume that you can just choose freely from the pool of unconfirmed txns and not, say, have to go through them in a specific order and to get to the ones with a fee you have to do the fee-less ass well.
Correct - miners can include any transactions they like. They
could choose to ignore the ones without fees, and this - or even ignoring transactions with low fees - has been a concern, but is generally suggested as being how the sort-of free market of mining works. Note that many miners will include zero-fee transactions right now anyway because there's very little reason for them not to*3.
* And here's where I get to explain all those asterisks.
*2/*3: There is
some reason not to include zero-fee transactions, or even to not include any transactions at all: it takes a small amount of time to do so. Every second wasted on doing this means a competing miner has a second more to mine a block without including all those transactions. That's one reason why sometimes you'll find blocks with only the coinbase transaction (the mining reward) and no point-to-point transactions at all.
*1: The corollary to that of course is high value transactions. If a miner is already working on a block with a particular set of transactions, but my transaction with the
BTC10 fee is too lucrative to pass up, they may well stop working on the block header they were working on, include my header, and try to mine that. There's some time lost in having to include it, having to restart mining processes, but such a high fee might be worth the risk.
Is it because sha is only applied to headers and therefore the number of txns in a block is not harmful in the mining process, ergo you support the entire network by confirming a lot of transactions?
Yep, see above. I had to double-check, but the main limit is the total size of the transactions in bytes combined, and that limit is something that's been under discussion for a while now, especially as the number of transactions (and therefore potential combined size) increases.
currently, according to blockchain.info the number of unconf. txns if around 1500 and is then reduced by several hundreds. why is this number usually over 1k and not like: 700..10, 670...20, 750...40, as i would have assumed?
Transactions are a bit like the tides with ebb and flow. When I checked a bit earlier there were 3,000-odd unconfirmed transactions, but alt-tabbing back over now, I see a block that included 1400 of them.
You might well at some point see 'unconfirmed transactions: 0', as long as miners include enough transactions. Right now there's 2,000 unconfirmed, and if another block like 350092 (2184 transactions included) were to happen, that would be the case.
There might also be some unconfirmed transactions that will never get included (let alone confirmed) which may be part of that counter - that'd be something for blockchain.info to explain in detail.
