This is often a trial-by-error business. Sometimes you have to try a few things to get optimum results, even if it means losing business in the short term. It’s a process that never ends for a host.
Are you booked pretty solid this summer? Raising your prices probably won’t help if your bookings are slim, unless you try to market to a more affluent guest base, I.e. stressing luxury components rather than ping-pong tables and fire pits.
If things are going well booking-wise, you might want to try the following:
Check other properties comparable to yours that are a similar distance to Downtown Austin. I am familiar with your city and area, and you are priced quite reasonably for being able to host 8 people. On the other hand, Austin is one of the places in the country that is over-saturated right now, so that might affect how much play you have in your prices. If others are priced the same or higher, and also seem t be booked well, you might try raising your price.
Try not to compromise on your policies. Allowing pets is opening up a significant array of potential issues. Plus, it sets a precedent for guests being able to push other boundaries you might have set. Be kind but firm.
On the sticky subject of reviews, as @Mike-And-Jane0 implied, it is much better to be honest, for your sake, the guest’s sake and for the protection of the rest of the host community. Guests can’t read your review until they have written one or 14 days has passed, so you don’t have to worry about a retaliatory review based on your review. If you complain and the guest leaves you a bad review, you can briefly and professionally respond to the review, stating actual facts and reasoning. This is actually an opportunity to sell yourself a bit and display your hospitality skills. Being honest also helps the guest improve, because future hosts are going to see your review, and the guest is going to have a hard time booking future properties. This results in guest behavior improvement.
Although I’ve never been through the process myself, I think there is a way to block guests from attempting to book with you in the future, but I’m not sure there is a way to do it online. There is a “Would you book this guest again” question asked during the review process, but I have heard some hosts say that it doesn’t block the guest. Best to call Airbnb.
Remember that you can try different things, some of which will trigger improvements, and some of which won’t. Nothing is irreversible, however - you can always stick your toe in the water and yank it back out if it’s too hot. Being a host is an ongoing, iterative process.