They were denied and told their old disposal would be removed, they offered to pay for a new disposal, they were told that they could but they couldn't install it or have it installed, but the LL/owner would install it and from that point on take possession of it.So basically they either have to pay more to live in a house with the same functionality or pay the same to live in a house with less functionality.

Then you go to court, the judge can grant you possession (or not, depending on the situation), and you can get your Writ and schedule a set out with the sheriff.

As far as abandonment is concerned - if she told me that she found a new place ( but did not specifically specify that she was moved into it) - is that the same as saying " i will not be back" even if she has left personal possessions such vacume cleaner, coffee pot, several photographs still pinned to the wall and nick knacs about the house.

I went this route with the thought my possessions would be in good hands and I could have time between tenants to stay in the house when I visited my siblings in the area.

For example, if you were the seller and needed 6+ months to move out of your home for whatever reason - would you be ok with taking a lower offer if the buyer said they're flexible with closing and can give you a few months post-possession?

Your standard contract, if it calls for possession at closing, means the seller is out Before/at the time of,  the actual closing.  

I know that in the city now, magistrates are hearing these cases and in some cases granting possession.

Included will be the delivery of good title at settlement, having a title search and insured title, who pays for settlement costs, when possession is given, inspection periods, insurance provisions, other issues including actions to be taken in the event the parties breach or fail to perform and disclosures required.

I see three paths to gain possession and please clarify if there is another way. 

(h) You must receive written consent from us to possess a clothes washer, dryer or portable dishwasher inside your unit.