The Hero's Path

Is it worth buying more paladins before the unicorn and demons, or should I buy some other troops?

It's difficult to rely solely on paladins; upgrading crossbowmen as an alternative, along with the blessing from the monks, can be very helpful in the early stages.

Is it worth buying alchemy research immediately in the first castle?

It's not very expensive, and it can be useful at any time. Moreover, it needs to be developed gradually, meaning initially creating many weak potions, possibly even just to sell, so that the skill improves and starts to really help.
But not in the first castle, but in the second. You can't buy it right at the start.
And it would be a shame not to try out such a feature that you won't find in other maps.