Is it possible for me to be able to partake in things like TOSH and the other high level dungeons while being free to play?
In short - NO.
There is an option of grinding 8 to 12 hours per day which, eventually, could get you there. When you get to the point where you can quickly solo DoD (8 to 10 months into your grinding), things will get faster as you will be getting 5 mementos (not 7 because some stats you will have to sell to get puries) and 2 shells stats every week. 7 stats is about 40 mil right now so you could get OD gch piece every week at that point. To plus it to 12 would take another week, to get runes slots and runes, another week and then about 2 to 4 weeks to get good stats (depends on your luck). That comes to be about month/month and a half of 8 hours per day grinding for one great piece of gear you will need for ToSH. You could skip building gear and just buy it already stated and plused but that does not decrease the cost. In just a bit over 2 years you will be all set for the ToSH.
Few notes: I focused on the stats as they seem more steady source of income than anything else. Also, i considered the latest prices for the gear in AH. Since they removed ABLs, GCH gear is no longer available there but I assume at some point it will be again. Also, I did not offer as an option proper progression through all of the instances, as that would require dedicated group of people doing the same thing and being in this for the long run and I did not see them around. Also, running instances instead of grinding would be more fun, but it would slow progression a lot as it does cost too and you will still need proper stats eventually and there is no much gold to be made in the lower level instances (there is almost nothing right now in the hos/hodl/zs you could sell as it is - even if you would have ABLs).