How to use 一点儿 and 一下儿？ They both mean "a little bit" according to the dictionary. However, does the meaning both apply in V and Adjective? 

i.e. 
V + 一点儿/一下儿 (吃一点儿/吃一下儿，快一点儿/快一下儿) and 
Adj + 一点儿/一下儿 (美一点儿/美一下儿, 好一点儿，好一下儿) 

In the above examples, my native Chinese friend told me she rarely or even never heard 快一下儿，美一下儿，好一下儿. So my question: is there any difference of meaning and usage between 一点儿 and 一下儿?
一下 is a little (time.) This makes it mostly after a verb. I believe the two most common uses are:


等一下 (děng yíxià) Wait a bit
休息一下 (xiūxi yíxià) Rest a little while


It is also used as a numeral classifier after a verb, indicating an attempt or act:


看一下 (kàn yíxià) have a look
试一下 (shì yíxià) have a try


一点（儿） is a little (quantity.) Some common uses are:


我会说一点儿。 (Wǒ huì shuō yìdiǎnr.) I can speak a little bit.
快（一）点儿！(Kuài diǎnr!) Hurry up (a bit!)


It also gets used before a negative expression (or after 这么 or 那么,) meaning "(not) even a little bit"


一点儿也不方便。 (Yìdiǎnr yě bù fāngbiàn.) Not even a little bit convenient.
这事我就知道那么一点儿。 (Zhè shì wǒ jiù zhīdao nàme yìdiǎnr.) That's all I know about it.


Contrast this to 有（一）点（儿） which is a little (degree.) Examples:


我有点儿累。 (Wǒ yǒudiǎnr lèi.) I'm a bit tired.
第十一课有点儿难 (Dì shí yī kè yǒudiǎnr nán.) Lesson 11 is a bit difficult.