You should not have to shoulder 100% of the responsibility of upkeep of the house. He lives there, too. He can help with the basics, starting with cleaning up behind himself. If he spills it, he wipes it up, if he dirities it, he cleans it up or puts it in washer. Not hard, and even those two things done daily, would make life so much easier on you.
A baby is a lot of work. You DO have a job, it is called being a full time mother, maid, cook, wife.
Get a neighbour to show you how to run the mower. If you can drive a car, most likely you can do this. Safety issue my foot! I mean, yes, be safe, but driving a car can be a safety issue, too. It isn't the vehicle (or mower) it is the person using it. Follow simple rules: Wear goggles, long sleeves and long pants for debris that may be there, and earphones for the noise. Really, anything over 80db should have protection on your ears. Mowers can be well over 100db and rival a jet engine's roar of 160db. Keep hydrated. Take breaks if your yard is big, or if you need water. If there is no one that knows how, google it. See if you cannot get a neighbour to watch your little angel for the time it takes to mow your yard. Return the favour with a homemade dessert, or babysitting for them when they go grocery shopping, or something like that.
Vinegar and water can be amazing cleaners. Look for cleaners that are friendly to your system online, see if you can make your own versions at home. I bet you can. For a lot less, too.
Tell him if he cannot help clean, then he can make his own meals, do his own laundry, too. You don't ask for much, you are not asking him to take all the responsibility, just clean up after himself, and maybe a little more when he sees you are exhausted, or struggling.
I hope I have not been too harsh, but sometimes, the working parent thinks once they walk in the door after their shift, their job is done. Some also think since you don't punch a timeclock, yours never starts. Neither statement is true. His job at his employment may be done, but he has a family that needs him to help out, a baby that needs to bond with daddy, a wife that needs an extra pair of hands when hers are weary from all the day's chores of cleaning and taking care of baby. Like I said, you are not asking him to take 100% of the load, just enough to help you.
Good luck.