Plumbing woes

Oh dear. Yesterday I decided to tackle a project that was seemingly simple. I read up on it, went to the hardware store and got the necessary tools, and now it's an unfinished mess, one that might require a plumber! It is not my fault, however, that it might require a plumber, but rather the fact that the fixtures in this house are OLD.

Both of our bathroom faucets have been leaking. In the downstairs bathroom, the faucet drips. It doesn't drip too fast, but fast enough that we're wasting water and money. To fix a drip like that, you need to disassemble the knobs, take out the spindle, and replace a washer. This is as far as I got:
I realize it's hard to see what's going on here, but I've gotten the decorative cover off, the knob off, and even two of the pieces holding the spindle in place. The spindle will not come out. It is so corroded and old that I cannot remove it. Sigh.

Upstairs, we have a slightly different problem. Instead of dripping out the spout, this faucet leaks around the handles. There's a constant pool of water. Although small, it's enough that we're tired of it always being wet on our counter. This problem requires replacing the o-ring on the spindle. So, I unscrew the handles, and get stuck at this point. Under the curves of the handles here, there's a small hole where you insert a hex key to unscrew the mechanism. Nick eventually got one of them off (I couldn't get it to budge), but the other one is too corroded to even unscrew the hex screw. You can see in this picture below just how corroded the hnadle on the left is:
So, Nick heads to the hardware store (I'd already been twice. It's a good thing it's only a block and a half away), talks with an employee about our situation, and we finally settle on buying a new faucet to just replace this old and broken one. I've already had the water turned off to the sink the whole day, so Nick goes after it uninstalling the water hoses. Bella is very curious about all of this:
Water hoses are unattached, but we run in to what has become an insurmountable problem as of right now. There's a weird crossbar attached to the nut on each handle post:
There is nothing explaining how to remove this in our lovely home repair book. I google like mad trying to find a solution. Those white brackets aren't screwed into the counter, as they appear to be, so we can't get it off that way. We can't remove the nuts, or even attempt to loosen them, with that crossbar in the way. If it originally slid into place, it is now too corroded to simply remove. So, I'm headed back to the hardware store when it opens with this picture an to beg them for some easy solution. We do not want to hire a plumber to install a faucet. We don't even know who we'd call since no plumbers have ever been recommended to us.

After reaching this impasse last night, I reassembled the downstairs faucet and turned the water back on so we'd have at least one working sink. Now it's dripping even more than it was! This is a problem we need to solve immediately, since we have a guest arriving tomorrow. Nick has volleyball tonight, so won't even be home. Hopefully my trip to the hardware store results in a solution that doesn't involve hiring a plumber, and I'll be buying another new fixture for our super-drippy downstairs faucet. I'll be tackling the rest of the uninstall/install on my own. I guess that's appropriate since I started this mess in the first place. I've learned a lesson here, though. What might seem like an easy project will not actually be easy. I'll be saving similar projects for the weekend in the future.