As for little, the difference is this: if you say that I helped you a little, then what I said was useful, but did not completely solve the problem. If you say that I helped you little, then I did not help you enough, you would like me to help you more. In so far I would say that "a little" = немного, "little" = недостаточно.

"Little" as an adjective means малый, маленький and various others. Whether it then has an article or not depends on the grammatical role the noun it belongs to fulfils.

"Little" in the dictionary: little : Englisch » Russisch : PONS.eu

"Firm" as an adjective means a lot of different things: firm : Englisch » Russisch : PONS.eu

Фирма is "firm" only as a noun. And the noun again can take articles depending on the grammatical context.