Can we enter Thailand on an exempt visa and convert it to an ED visa while traveling with our cat?

Thanks for this group -- it's really helpful! I have a couple questions and would love advice.

Background:

My husband and I have been to Thailand several times and would like to stay longer. We returned from our last trip on December 2018. We are both over 50, US citizens and we are considering the Retirement Visa (and meet all requirements), but want to see if we truly want to relocate long-term first.

Questions:

1. Can we enter Thailand on an exempt visa, get paperwork from our Muay Thai school (accredited school and they are doing our paperwork in advance so we will have it within 10 days of arrival in Thailand) -- and then head to Laos to convert it to an ED visa (within the 30 days)? Or do we need to first have a single or multiple entry tourist visa from the US in order to convert to the ED visa in Laos?

2. We're traveling with our cat (pure breed, we have all the details handled). We will also have proof of onward travel before our 30 days for the exempt visa. Does anyone think having a cat with us will be a problem on an exempt visa? (As in, raise red flags for an exempt visa.)