When you write "given my mental state", I sort of wonder how bad it is and if it got worse what some of the ramifications could be. Consider the following:

You write that you want a pet (a dog) as an emotional support animal. I am the first person to say that having a pet has been very valuable for me over the years. I have loved them all, and they have all indeed given me emotional support. However, there have been times when I was so ill that I could not properly take care of them, especially my first two parrots during some periods. When I couldn't, my husband did, just like if I had had kids he would have taken care of them when I couldn't. It's not a one-way street with pets. They need basic care, too. They can also become distressed when they see their primary owner become distressed. If/when you become unable to properly care for a pet, is there someone who can help?

Very soon after marrying my husband he wanted us to move from the apartment we had to a house we would buy. I was not thrilled with the idea at first, but he was very insistent. He made great points. A house allowed greater freedom, and it was an investment. We did pick a house that was in very good shape and required next to no renovations at the time. The house we picked was also a townhouse that was part of an association. The association handled things like lawn upkeep and snow removal, garbage pickup, and house exterior upkeep. All of these things need to be added to the price of a house as monthly fees, as does homeowners insurance, and property taxes. Perhaps where you live there is no snow, but are there other issues that could increase homeowner's insurance, like the dire need for flood or earthquake insurance, etc.?

Buying a house WAS a great investment in the end for us. We were lucky that we bought at a time when I was working full-time and doing very well. That would not have been possible in recent years. We have lots of equity, however, it is getting hard for us to afford living where we do on just my husband's salary and my disability. We live in a high property tax area (outrageously high), and other expenses are high, as well. I've been on disability for a long time. Living in the US, my medical bills have been outrageous over the years because my illness was severe. It's not like living in Europe where health insurance is not a worry and people don't go bankrupt when they get severely ill.

I do not wish to discourage you from buying a house, but just wanted to mention how many factors there are to consider in this. Definitely a house that may need renovations/fixing can really add to the burden of home ownership.