air purifier questions - windows, dog hair

Hi, I'm considering an air purifier as I wake up with blocked sinuses and headaches often. A friend suggested trying out an air purifier - I know it won't be a magic solution, but I'm open to trying and seeing if it helps. I'm in a studio apartment. A few questions:

• I like to have my windows open during the day. Will this make it pointless to run an air purifier at night? Basically, I'm not sure how quickly air purifiers act and if it's like a humidifier where you really need to build up humidity over time.
• I have a dog who sheds immensely. What will this mean for the air purifier? Will it render it useless, or will it just mean needing to replace the filter more often? I know I still need to vacuum and that the air purifier isn't going to help with visible dust/fur - my concern is more if I might "overwhelm" the air purifier, I guess?
• I'm looking at Levoit and Coway after reading a few guides. Would love any thoughts!

Thanks in advance for any help!