Question about moving my HVAC Furnace

[Tidybuoy]

I'm preparing for next year's kitchen remodel and phase I of the project requires me to move my Heater from one closet to another. Fortunately, the destination room is close to all ducting so this will be relatively easy.

My issues:
1) the current system has a terrible filtering system. To change the two filters, you open up the lower section of the furnace and the filters are there - but done very shoddy and air can pass around the filters somewhat.

2) I was thinking that befor moving the furnace, I could build a 6" tall base that it will sit on and I could incorporate a drawer that holds the filter. Does that seem like a good idea or am I missing something?

3) My plan is to empty the destination closet and then get the intake and output ducting complete before moving. Intake is from the floor and output is into the attic. Before I call an airconditioning mechanic, what type of cost should I be expecting to disconnect, move, and re-charge the system. I want to minimize the cost by doing some of the upfront work myself.

Below are some pictures of my current Furnace. A new one is not in the current plan but maybe in the next few years. The current output ducting is very crude as I think this was setup for an older heater (house is 70 years old).

Sorry for the sideways photos - I have yet to figure out how to get my iphone to do this correctly.