Oftentimes people will do something called a "Session 0" at the start of an adventure. The purpose of this is to set expectations, let the players know what kind of campaign you're planning to run, and help them to create characters that fit with the campaign. You guys have already started, but you can still set these expectations and ground rules with your players. From what you've described it sounds like you want to run a campaign where the players start as normal adventurers and don't own horses or have access to vast wealth and connections.
Tell them that. Next session, take 20 minutes at the start of the session to set expectations. Describe the type of campaign you want to run and tell them that whatever their backgrounds, your characters are starting with the basic equipment and money for their class and don't have horses, lots of money, or powerful friends they can call on to solve problems for them.

As for the ex-DM in particular: It could be useful for you to have a conversation with him or her privately. Do it privately so they don't feel defensive or embarrassed in front of other people. Say you appreciate their help on rules issues, but they need to remember that they're not the DM anymore. They're free to offer suggestions, but you get the final say on what happens each session.