Can transactions scale sufficiently to compensate the diminishing block reward by fees? Today, the bitcoin network is secured by massive computing power, which is expended in the process of mining. The miners are rewarded by newly mined bitcoins, plus transactions fees.

The security of the network relies on the computing power being so expensive, that no one group can afford to gain 51% share.

Eventually, the amount of newly mined bitcoins will diminish, and the mining  reward by itself would cause mining to become unprofitable. 

Will the fees be able to pay the expenses for the huge expensive computing power?

I can see two extreme situations where it might work: 

a) either the transaction fees will be astronomical
b) or there will have to be huge amount of transactions


option a) is clearly not viable, which leaves option b).

What would this mean for the bitcoin economy?

Will bitcoin need to become "high-velocity" money?
How could that be enforced? And how would we prevent the blockchain growing out of control?

Is my analysis correct?
Or a combination of both. It looks like things like the Lightning Network will be there for micro payments and smaller amounts so that only larger transactions will be directly on the blockchain. For larger (and aggregate) transactions a bit more fee shouldn't be a problem (a). Also at some point the block size will likely be raised allowing for more transactions per block as well (b).

Either way your title is wrong: there is no post-mining world.