Hi, it depends on which project you're tackling. Ubuntu MATE's projects are not strict and leaves the creative control up to the discretion of the translator.
Does translation has to be literal?
It doesn't have to be word-for-word perfect, but should convey the same information.
Like in films and media, if the source text wouldn't come across right, it can be re-worded to have the same effect to your locale's audience.
The only 'issue' is that the strings in 
Welcome may "split" up in some places and could cause problems where there's different meanings in other languages.
On Transifex, I just translate a string and click on 'Save Translation' button and be done with it?
Yep.
When I translate, how long does it take for it to reflect in the main project (website or application etc).
Depends on the project. I maintain the website(s) and put in the guidelines that as a rule of thumb, I'll check every month and publish those with >50% translations. Other apps like Welcome and MATE packages usually update translations before a new release is out. That could be weeks or a few months.
You can preview translations yourself by building a copy of the application/project(s) and downloading the PO files.