No. Units are not available until the turn after they are recruited. e.g. If you recruit 2 light macemen on Turn 1, you will not be able to move them until Turn 2. You can move any units you still have from the last turn.
Caveat: The exception to this rule is mercenaries. You can hire mercenaries of any unit type available to you for twice the normal unit cost. Mercenary units are available on the turn you hire them and you can use them immediately. e.g. A mercenary band of spearmen would cost £250 to hire. A mercenary band of archers or light macemen would cost £1000 to hire. Upkeep costs are as for normal units.
NB: Any units that start a turn wounded (due to battle) will heal fully by the next turn. e.g. If on turn 0 you attacked with 4 macemen, won but two of them were wounded (leaving you with 2+2 macemen on Turn 1), those wounded macemen won't recover until the start of Turn 2.
Spoon: I'm going to give that offer a 50% chance of succeeding, if that's alright. If it fails, you can just use the money on spearmen as you said.
Also, if anyone wants to feel free to roleplay. It probably won't affect mechanics much (since combat is all in the engine and has clearly defined rules), but it'll be fun and I might take it into account when Random Events start happening. (I reckon I'll random event every 3-7 turns, depending on what's going on.)