Regarding your performance: Do you have reason to believe you will receive a bad performance review after 6 months? If not, then no news is good news, it means your manager has nothing interesting to say to you and you can expect to pass your performance review. If you do believe you have a problem that needs addressing, then you should ask your manager for feedback anytime. You can even say something like "Hey, I noticed we haven't been having any meetings about my performance during my probation period, I just wanted to check in and ask if there was anything you wanted to tell me". Your manager won't come back with "Why didn't you ask me this 2 months ago if you really wanted feedback" or anything like that, that's ridiculous. Your manager should be happy that you care to receive feedback and should give you whatever they have. And if they come back and say "no, nothing important", then just take that at face value, it means you're doing a good job and carry on.
Regarding the design reports, 2 months is a long time for a design report. Not sure why you thought you'd be given 5 months; an entire project can be done in 5 months, not just a design document. Was there a reason you thought you had until June to complete them, and then the deadline was changed on you? If not, then this is mostly your fault for not confirming the deadline properly. If you believe you can't get the reports done in 2 months, then you should explain to the interested parties why and what the problems are ASAP.
Regarding your coworker: It's possible it's jealousy, and the way you describe it it sounds like jealousy. He probably was hired as a developer, but wasn't able to perform, and resents you for surpassing him in his job, while he is stuck doing "grunt work". If your company has review meetings on Fridays (many companies do), then perhaps he had his review meeting where his boss (your boss presumably) told him he wasn't up to snuff, and he sees you surpassing him and he's jealous and angry at you for "taking his job". One way you can deal with it is ask him what happened to his programming, like, did he receive some bad feedback or something? If you understand the situation and you feel like you are comfortable with this, you can go to your manager and advocate for him, like "hey, Bob has been telling me he's uncomfortable with the work he's being given, can you help Bob out a bit?". If you have a project you are having trouble with and could use an extra hand, you could request him to join your team and help you out. It would likely make him feel better and more appreciated. It sounds like you might have a report with a tight deadline that could use an extra hand anyway ;-)