Not sure what their pet policy is. As the place seemed to be quite disorganized and not too smart at the front desk area, they might not even notice you're bringing your pet. OTH, you might want to avoid a nasty surprise on arrival so it's best to double check with them. It's best to do it in writing so that when the inevitable front desk confusion arises, you can just show them your email that says it's ok for you to bring said pet along. Good luck.
The library around the corner has sheltered parking with unlimited entry/exits in a 24hour period for $12. Upon entering, park your car, go to the machine and deposit money. It will print a receipt with a time on it; leave the receipt visible on you car dash. If you want to leave before you time expires, you are free to do so. Likewise if you return before your time expires, you can still use the same receipt (just make sure to pay for the next additional time you need.)
There is street parking directly in front, however it is just for unloading. There are other spots that have time restrictions, and cost money.
We parked our car at the library, and used bikes to get around the city. Around the corner there is a BIXI bike station that offers very cheap bicycle rentals per day (I think $5/day). There is a 30 minute time limit before you have to drop it in another BIXI bike station, but then can immediately take one out again for another 30 minutes.