If a soft fork is activated, why would someone be mining in "old format"? I'm assuming that you're talking about a miner/node that is using pre-Segwit software to mine, in which case it would get orphaned I believe (previous version block). If you're referring to something else, then please elaborate.
No, not necessarily. Firstly, version numbers are no longer used for signalling and "forking" as versionbits is now used. A block is still valid if it is in the non-segwit format and contains no segwit transactions. It is only invalid and thus rejected by segwit nodes if it contains segwit transactions but does not use the segwit serialization format or the have witness commitment in the coinbase.
I am still thinking what will be the best strategy for a miner, if he receives
a block in "old" format from non-upgraded node. Such blocks will transfer
much more faster through the network, because there will be no signature
checking operations.
Should the miner throw this block away because of lack of witness parts in transactions
or should it mine on top of this incomplete block?
If a node that supports segwit receives a block in non-witness format but contains witness transactions, it should reject it as invalid.
