I cannot find a wiring diagram for this unit so the below is not written in stone.
If you do not have it check in the control console they often store info in it.

Very strange that the hi-limit keeps blowing open, usually they blow closed and the thermal-cut-off blows.

Terminology is important here.
By hi-limit thermostat do you mean the one on the heater assembly?

If you just removed the 2 wires from a device then yo could not make a mistake as it does not make a difference which goes where.
An exception is the 4 wire cycling thermostat 2 wires are for the contacts the other 2 are for an internal heater.

Now for the terminology.
The blower assembly has a cycling (bias) thermostat and a thermal fuse.
The heater assembly has a hi-limit thermostat and a thermal cut-off (fuse)

The high limit should never open unless there is a problem with the unit. It is just a safety device.
The thermal cut-off is just a back up device for the hi-limit in case it's contacts should weld together.

Here is my standard blurb on this for a blown thermal cut-off, hope it helps.
Check the heating coil.
Unplug the unit and both wires to the coil.
Check it with a meter, should be around 10 to 12 ohms.
Then check from each side of the coil to the case/frame, both should be infinite ohms (open). If not the coil may have sagged or broken and is touching the case. This can cause it to run on high and the thermostats cannot regulate it so the thermal cut-off blows.

The hi-limit should have regulated the temperature so the fuse did not blow, that is why there is a new one with the thermal cut-off..
Note: That unless there is another problem in the unit the hi-limit should never have to open. It is just a safety device with the fuse being a backup safety device.

Just in case it is not a grounded element.
With all the below the high limit will also have to be replaced.
Check that the belt is OK.
Check the seals (drum etc) in the unit. The air is pulled over the heating coils, through the drum and pushed out the exhaust. So any large seal leak will pull in room air and the cycling thermostat on the blower will run the unit hot.
Check that the lint filter is not coated with fabric softener residue which greatly reduces air flow.
Check/clean your vent system.
Check/clean the blower wheel.

If all OK you may want to replace the cycling thermostat as it's contacts may not be opening (welded shut)

[COLOR="DarkRed"]Remember it ran for about 5 minutes after I replaced the thermal cut off the first time. But then checked each side to frame-got no meter reading? What does this/that mean? Is this a concern?[/COLOR]
No it should be an open (infinite ohms) so this looks good.
If grounded the unit could run for some time before it blew the thermal cut-off.
I will try to explain.
The Neutral/ground/machine frame is not part of the heater circuit.
The heater power is L1 to L2 which is 240 volts.
Depending how it breaks/grounds part of the element may be connected to either L1 or L2 and the case/Neutral so current flows all the time and the thermostats cannot regulate it so it can overheat.
I hope the above makes sense..

[COLOR="DarkRed"]I am going to replace the cycling thermostat and thermal cut off.[/COLOR]
I would also replace the hi-limit. Since the element looks good the hi-limit should have opened before the thermal cut-off blew.

Also check that the seals in the air flow path are OK.
Your cycling thermostat is on the blower and if it sees a cooler temperature then it will keep the heater energized.
From the air input to the blower the system is actually under a vacuum and can suck in cool (room) air if there is a leak.