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Hi, I'm Mike G and I'll be glad to assist you. Unfortunately you can't run the individual conductors outside of a conduit. You may use a cable of the same size, spliced at junction boxes at each end or from panel to outlet.
Unfortunately no. I use the Tyco Splices and only know that the #12 wire as being the largest
The video shows a violation of code. Splices an't be made on individual conductors in anything other than an underground installation.
That splice needs to be in either a junction box or underground to be code compliant. You have no alternative other than a cable replacement.
Had you considered a suspended ceiling. It would allow you to leave the wiring as it is and provide access to the junctions and anything else in the future.
If you replace it, current code requires it be a 4 wire. 2 hots, 1 neutral and a ground. The wire size for a range is #6. If it is a dryer the size is #10, 4 wire also.
A 2 pole 40 or 50 is acceptable for a range.
It will run when there is a demand from the t-stat. The T-stat will be on a 120V circuit to the furnace, which you may have turned off.