Q: Car having hard time turning on, when it does turn on it sounds like its struggling and is almost shaking.

After coming back from a 2 week vacation (the car was sitting in the parking lot) the battery was dead. I passed it off as the mix of cold weather& it sitting for so long. I got it jumped, drove the car for 30+ minutes & got the battery checked. Battery read as good. Few days pass & it died again, I left it alone came back to it & again tried the car, after some struggling with the car it turned on. Struggling = first attempt to turn on it wouldn't click like it was dead, tried again & the sound was like it just wasn't catching to turn on, third attempt it caught & turned on. Drove fine. Didn't use for a day & again it was dead, or wouldn't turn on. Went through the same struggle of it sounding like it just wasn't catching to turn on. Took to a different auto place & again battery was fine, they asked a few questions & said alternator was fine too. It recently started to shake & sound like a fan was working hard while driving, but doesn't die while driving, nor do lights dim.

My car has 135000 miles.
My car has an automatic transmission.