Manel, I am going to point out obvious points for your case, and I hope they help, I am not a dev, but I am looking at some of your questions and coming from my own experience with my answers, I hope they help.
Firstly, I would create a copy of your tiki, and use it to upgrade with first, before you upgrade directly to your live tiki. Since this is a company tiki we are talking about, a week or a month to make this happen in my opinion would be worth the effort. Even if you have to purchase a new domain for testing purposes only, and copy your live tiki to that domain, and then perform your upgrades. You could test it both ways, 14 - 15 - 18 and then perform the upgrade straight to 18. And see how it reacts.
As far as migrating your db tables and the added ones, if I am not mistaking tiki upgrades your existing db, and should not alter your existing added tables.
The theme, is your theme completely custom? Or is it modified from the native boot themes?
One thing I am certain of, is make a copy of your tiki, and experiment with upgrading the copy, and analyze how it reacts to the new changes. And if anything is out of place, you'll be able to address those issues in your copy, make note of them, and for your final upgrade, you'll know what is coming.
And I am sure you have backups galore since it's your company tiki right? Obviously back up before you do your final upgrade. I know you knew that.
Hope this helped just a little.