Also can't ever get to my gifts. Always opens my co-op. Sometimes I can literally only get to my gifts once or twice a day, because it just keeps opening the co-ops.
I found a few things that help with this.
First, it helps to follow certain paths into the Gift Box. There are probably a dozen ways this could work, but what I do is click somewhere on the webpage, but out of the Flash frame, and don't click anything that does anything. What you're doing is (in web developer-speak) setting the browser's "focus" on something that does
nothing. That really helps. Then, I always come at the Gift Box hotspot laterally (from the side) -- not from the top! I also tend to move my mouse cursor over the Market hotspot first, and therefore finally come at the Gift Box hotspot from the left, instead of directly from the right. I don't know why that seems to work better for me, but it does. So now you've got the pop-up menu exposed, move the mouse cursor up to select the Gift Box, but don't select the picture... only select the word Gifts, and only when it has a yellow shadow behind it.
It sounds complicated, but you do it a few times and it become second-nature and not in anyway an annoyance. It's like learning yoga... once you learn the proper way to come into a posture, it is harder to do it wrong, than it is to do it correctly.
This doesn't work for me 100% of the time, but it has increased the amount of time I get reliably into the Gift Box, by a large amount.
At first, I thought that this meant that there was no way to make it easier to get into the Gift Box when you have the game full-screen. That's actually not the case. The whole bit about clicking somewhere off the Flash frame to put the browser's focus on something that does nothing doesn't seem to be necessary when you're in full-screen. I seem to be able to very reliably get into the Gift Box by following the mouse actions I described above (starting from coming at the Gift Box hotspot laterally). I've only been focusing on it for a few days, and it seems to work almost as well.
I was so hooked on Farmville a few months ago, but I've really started to step away. It's just not worth the aggrevation.
I think I look at it differently... Accepting FarmVille as it is, is it better than what I'd be doing instead (playing some other mindless game that isn't as appealing overall, and probably has its own foibles to get frustrated about, if we let it)? If you've already got something better to do, then obviously that's what you should be doing! But if it is just a matter of it not being perfect, I've found that accepting its imperfections still leaves me with a game that is more fun than what I'd be doing otherwise.