Also, what "threw you off" is normal for generic light switches. Lights need a complete circuit to turn on. You use a basic light switch to turn on/off just the black hot wire, which breaks the complete circuit to the light. There is no need to also break the white wire, so that's why they're all connected, just passing the neutrals through the j-box out to the lights. The switches should have been grounded for safety, but doesn't affect the operation of them.

Sounds like you wired it backwards. If you're leaving the switch on the right, you want the jumper to connect to the black on the dimmer (left). The red wire from the dimmer should connect to the black romex wire going out of the box (on the left). That black wire goes to your light. Before you actually make that connection, I'd suggest fully disconnecting the switch on the left, capping the jumper, turn power back on, and check the black romex wire on the left with your meter to make sure it is NOT hot. If that wire is dead, with the breaker turned on, then it is most likely the hot wire to your fixture and is the "switched" hot wire.

Going to start with the usual disclaimer that if you're asking these fundamental wiring questions, you should call an electrician.

But, if you want to learn and are willing to take the risk, you should start off with buying a digital multi-meter, so you can check which specific wires are "hot". What you're likely dealing with in this box is a single "hot" wire (black) that is bringing power into the switch on the right. From there, you have a black "switched hot" going to a light from your switch on the right. You also have a jumper making a daisy-chained connection to the switch on the left (which is how it gets power). Then from that switch, you have another "switched" hot that goes out to a light. Based on that simple wiring, I'd wager you do not have a remote 3-way switch tied to either of these.

As for your new dimmer, if you look in your instruction manual, a standard single-pole dimmer connection has you connect incoming "hot" to the black wire, the dimmed output to your light is the red wire, the green goes to ground, and the red striped wire is capped. Because both dimmers have black leads, you could tie them together and connect to the incoming hot, if the leads reach.

If you don't have much experience with wiring, I'd again advise a professional, but there are many YouTube videos on how to properly use wire-nuts or lever nuts to make these multi-wire connections. Your scenario here is very basic lighting connections. Make sure to turn off the breaker before touching anything in the junction box. If you need to check which wires are live with the multi-meter, turn off power, disconnect the wire nuts as needed and make sure there is no chance for a short, turn on power, check with meter, turn off power again.

I had an identical situation when I remodeled my kitchen. Even with the old dishwasher under it, the counter still flexed a little in that spot. Since I was installing quartz counters to replace the laminate counters, I didn't want it to be unsupported because quartz was likely to crack instead of flex.

My solution was a large L-bracket that extended from the wall near the dishwasher/oven seam. I ordered a 22" Universal countertop support bracket from CenterLine Brackets (link below). Quality was top-notch, but it did require me to cut open the wall and install 2x8 blocking between studs to give it something sturdy to mount to. I also had to precisely mount it to be level with the rest of the cabinet framing so the countertop would be level. It gave me just enough clearance on the back of my new dishwasher to put the dishwasher in without hitting the bracket.

https://www.countertopbracket.com/Countertop-Supports-Floating-Inside-Wall-Mount-p/fwm.htm

Here's a photo of it installed https://imgur.com/a/jao7Cmm