shop 쇼핑하 하 → 해::::::::::: 쇼핑해 

Actually, I'd like to point out that 해 comes from 하+여, but a lot of resources will tell you "just to change 하 to 해" because the result is always the same. But when you are reading, you may come across 하여, so know that is actually 하+여=해.

ㅐ+ㅓ = ㅐ, so yes, it applies to all verbs.

ex. 돈을 내다 -> 돈을 내요.

Again, when reading, there are times you will see uncontracted forms of verbs.

It also has changed like 친구 만나. The verb 만나다 became 만나 in this case. 

Samething, 문자보내다 here, the verb is 보내다 became 보내. 

다 is simly removed out of the verb When you make a simple, casual, and informal sentence of as an order, request, suggestion and etc. 

but when it ends with 하다, 하 should be changed as 해. 

테니스해, 수영해, 쇼핑해, 편지해, 문자해, etc.