This way I can get the snowball rolling even if it technically is not the best use of my money at this point.

Ask for seller's tax bill and title to provide tax rolls, call the county for the add ons... 

Or would It be better to just sell and roll the equity into our next home?

If not, you need to start saving and get prepared for that and roll into a month to month VS a new term so when you have enough saved up you can give notice.

Let's say I want a house for cheap, so I decide to look for pre-foreclosures.

Sounds like overall you have the right attitude, use the challenges as a way to learn, and keep the train rolling!  

Now, when you go to REFINANCE that property, you will also receive 75% or maybe 80% of the ARV on the refinance step (basically enough to roll in closing costs at 80%). 

Is it good practice to open HELOC when I received pre-approval letter form lenders for new Home purchase?

What you should do is call this place as an interested buyer and ask for a few things to validate the numbers you're putting into the model, such as their rent rolls (you will see if collectability or vacancy is an issue), historical costs and income statement if available, and any large capital expenditures that are coming up.

We are at the point where we want to get pre-approved.