I've been trying to mix some items like Icebomb and Firebomb (Canned Elemental)but nothing happens. Yet I've read somewhere that this is possible. I figured that the Alchemist is like a Gadgeteer which needs a required level to succeed. Is this true? Please tell me some combinations and their Alchemy level to succeed.

You got it: mixing higher-level stuff requires higher Alchemy skill levels. Here are a few of the mixes I've found handy, either to make something useful or to make money by buying ingredients and mixing them, then selling the result for a lot more than the ingredients cost:

Alch 15: Light Heal + Moderate Heal -> Heavy Heal
Alch 30: Light Heal + Poison Reduction -> Cure Poison
Alch 40: Heavy Heal + Cure Light Condition -> Cure Disease
Alch 50: Heavy Heal + Cure Disease -> Renewal
Alch 50: Sneeze Powder + Flash Powder -> Pandemonium Powder
Alch 75: Dust of Dessication + Concussion Powder -> Skeleton Powder
Alch 95: Ice Bomb + Fire Bomb -> Canned Elemental

In particular, the 2 x Light Heal + 2 x Moderate Heal + Cure Light Condition -> Renewal is a good moneymaker, producing a net profit of over 3K per bottle.

Also, mixing stuff raises your Alchemy skill rapidly--faster than casting combat spells, in fact.