Be careful as some abbreviations have more than one meaning. 

lol = lots of love or laugh out loud. 

I've seen students have the following text conversation...

Student 1: My cat died last night.

Student 2: Oh no. lol.

Yes aberrations used by native English speakers can be very confusing for someone who is trying to learn English. To answer your first question "BTW" means 'by the way'.

Some other aberrations that are commonly used in everyday English are as follows...

(V.I.P) - Very Important Person. "He is a V.I.P we must take care of him."

(24 / 7) - 24 hours, 7 days a week. "I work 24 / 7, and I am very tired"

(R.I.P) - Rest in peace " My mother died last year, R.I.P"

(D.O.A) - Dead on arrival "The man who crashed his car was D.O.A at the hospital.

(D.O.B) - Date of birth " They asked me for my D.O.B when I registered on Facebook"

(A.T.M) - Automatic Teller Machine "I use the A.T.M to get money out of my account"

I hope this helps you.

Here are a few common ones off the top of my head:

irl - in real life

brb - be right back

ikr  - I know right

wtf - what the fuck

iirc - if I recall correctly

lol - laughing out loud

lmao - laughing my ass off

g2g - got to go

fyi - for your information

ty - thank you

Here are two threads on abbreviation/acronym--

https://www.italki.com/discussion/131437

https://www.italki.com/discussion/130935

Hope this helps!

Enjoy learning some crazy acronym:)