I've made a computer go engine and I need to test it online.
Is there any way to connect it to Tygem directly?

Not that I've ever seen. It's why a lot of authors use KGS to test it out.

Obviously, there is. Do you think that every (B) [bot] player on Tygem is handled manually?

Tygem simply might provide service for bot owners.
For example you can create bot account on KGS for free games - with any bots.
But you can't play ranked games.
To play ranked games for bot - you should be developer of this bot. And this can be done only via e-mailing with KGS administration.
So believe it's the same for tygem/fox.

Thank you for your information.