OT extension question about base-OT
How can I implement $\kappa\times OT_\kappa$?

You pick your favourite public-key based OT protocol. Usually these will allow you to input a choice bit and learn exactly one out of two $r$ bit values in return.
Then you input two random $r$-bit values into that protocol and use the $\kappa$-bit hashes as output. This is your $OT_\kappa$ protocol.
Next, you execute $\kappa$ (e.g. 128) instances of that protocol in parallel with independently random choice bits. This is what is usually done for these OT-extension protocols.
As for building a $OT_\kappa$ from $OT_1$s, you can indeed run $\kappa$ such protocols in parallel with the same choice bit (which is only immediately secure in a semi-honest model).