Tiling first then cabinets? Trying to understand the sequence

First this is the best site I've ever come across and will never consider big box store advice again! :no: But I'm trying to tie all the pieces together from everything I've read here and have some questions before I tile my kitchen...

1) Many of you mention tiling before putting down cabinets. But I will be installing an island with a gas cooktop and was planning to lay 2x4s flat on top of the hardiboard as a way to anchor the island cabinet to the floor. So my thought was to hardi the entire kitchen, install the cabinets, then tile. But what if I went with Ditra (which most of you like)... I would be nailing into the Ditra when I lay the 2x4s. Won't that defeat the purpose of the floating floor idea? (BTW, kitchen area is 340 sq ft and the island will take up 10 sq ft)

2) I also read here that you should never ever put SLC down first. So, you guys are saying I should put down the Ditra or Hardi first and then the leveling compound? I have some spots in the kitchen (near the walls) where the 3/4" plywood slopes down a good 1/2".

3) this is actually a follow up to an older post I had... JazMan and others told me my deflection was close and that I needed to shore up the joists. I was going to add a wall down in my basement to shorten the span, but after some thought, I think I'll just reinforce the joists as another solution which many of you suggested. But I have a dumb question to ask about that process, so I apologize ahead of time...
a) If I use angle iron or a 2x4 as a sister, do I place them along the top or bottom of my 2x10 joist?
b) If I just nailed a 2x4 flat to the bottom of the joist do I go the full span of the joist? Also, do I need to consider running any 2x4s perpendicular to the floor joists? Right now the joists have these criss-crossed pieces of wood between each joist, which I guess is to prevent the joist from flexing one way or another.

Again, I thank you all for your help and expertise.

Sorry for all these questions... I actually went to two tile stores yesterday and asked these questions to their installers and got different answers. And when I asked about dealing with the deflection (which way to lay the 2x4s as suggested by JazMan, they just pointed me to the Ditra). very frustrating.