Firer bomb does more than twice the base damage of entangle and acid bomb(It's water the description is wrong) does more than triple. Most undead are weak to fire and light and resist pretty much everything else so firebomb will clear an entire line of undead in one hit and finish a battle in two rounds with only half the mobs even attacking whereas entangle can take 3 or four with everything beating on you the whole time. Fire bomb is mostly useful in crypts and bandit forts(ballistae) whilst acid bomb really come in handy for ancient jungles and fire pits where it can one hit beetles, ifrits, and salamanders. The most profitable dungeon to farm off the bat is usually bandit dens which favor aoe earth attacks but alchemist will make your life a lot easier if you have to kill undead for quests and you are using a caster type.
As for Assasin you can get it before day 21 but it's a serious stretch. In my other day 14 post the class order for my fairy caster was attendant> apprentice> druid> thief> rouge> alchemist> harlot> succubus> archer> fighter. The only equipment I gave her was the starting mage staff for siding with mages and that should be fine until you get a lvl 3 alchemist. I don't recommend using your MC for combat, it's better to kill everything with your free recruit for a number of reasons. I can go into more depth if you're interested. Using a single character for combat rather than a larger team will result in a more powerful combat team faster since it's the synergy of many classes that makes you strong and xp is split amongst all participants. It's also handy for power leveling laborers later on for much the same reason.