If you are in Bangkok you’d need to have done a 90 day report in order to get a residence certificate. In other provinces you don’t have that requirement.
That’s going to be up to your airline. Some require you to have a ticket out of Thailand within the number of days allowed by your entry scheme, regardless that you could get an extension. Some don’t seem to care.
You probably need to speak to a local dog company to see what the requirements are. Soi Dog in Phuket will be able to help. If in another area check out their places.
There is no technical limit on the number of times you can enter Thailand using visa exempt entries. Twice in a year isn’t going to raise any red flags for immigration.
They may question you on the 3rd or 10th entry. But as it’s up to the immigration officer no one can say precisely.
What passport country are you from? There is no such thing as a 58 day visa.
You are probably referring to a tourist visa which is 60 days. The day you enter is day 1 regardless of the time you enter so you get stamped in for 59 remaining days.
However many countries now can enter with no visa and get a stamp for free for 60 days (59 extra days).
For many there is no point in paying for a tourist visa.