Did your landlord file an N12 and pay you the appropriate one month's rent for having to move out? This would be money in addition to you not paying your last month's rent due to first/last being paid when you signed your lease.
Based on your claim you need to be out in 2 months, I am guessing your lease is being terminated illegally. Filing an N12 to evict a tenant for landlord's use requires the landlord to provide 90 days notice, not 60.
If you did not receive an N12, YOU DO NOT HAVE TO MOVE and are LEGALLY entitled to stay.
If they file the N12 tomorrow, you will have 90 days to leave your rental based on the day of the month you took possession of your rental.
For example, if you moved in on the 1st of a month and they hand you an N12 tomorrow, you are legally entitled to stay in your unit for 90 days as of March 1st (aka: the place is still yours if you want it until May 31st).
Once your landlord files, you only have to give them a couple of weeks notice to end your lease and move when you find another place, not the usual 60 days. They still have to pay you your one month's compensation, plus reimburse or return pro-rated your last month's rent if it was not used to cover your last month in the unit.
Simply telling you they will move in is illegal. Ontario law requires the filing of an N12. The N12 should be the standard government form which also informs you of your rights.
Onto the pets situation, pets like dogs and cats are generally allowed in traditional rentals, provided they do not destroy the place or cause noise issues (for which you can be evicted). There is no legal way for a landlord to prevent your dog living with you in a future rental.
If you are finding push-back from potential new landlords, try purpose-built rental buildings or simply not mentioning it without being asked. The Ontario Residential Tenancies Act does not allow a landlord to kick you out or place restrictions on your having a pet. Any ad posted for "no pets" or a "pets exclusion" clause on your lease are not legal and you cannot be evicted nor charged for having a pet unless said pet is destroying landlord property or causing a nuisance to other tenants.
Sucks to say, but in this rental market, probably smarter to avoid saying you have a pet, especially if you won't be living in a building where the landlord lives as well (i.e. condo or rental building).
TL;DR: no N12, you're good until May 31st. "No pets" clauses are illegal. Check the LTB site for more information. Know your rights!