Yes, That LED strip will work if you want 12v and 5mm.
Yes, That dimmer will work for your selected 12v LED strip. Yes, you can use it with the stock power supply too (though I can only assume the connectors are compatible in size since I don't have that info).
Problem here. The connectors you selected are for 8mm wide strip lights. You need 5mm wide if you plan to use the strip light you noted above. I looked quickly on amazon and don't see an equivalent connector kit for 5mm (you can check further). You need these same contents but for 5mm strips. Specifically, the connectors that attach to the strip won't fit. You need 1) minimum two strip connectors that fit 5mm strips on one end and have a female DC barrel connector on other end , and 2) One DC power splitter that splits from one female barrel connector to two male barrel connectors. You could explore more to see if you can find these items.
Alternatively, you could change to the 8mm LED strip light, and then keep the connector kit you selected which would then be compatible.
I'm assuming these DC power barrel connectors all fit compatibly. I didn't use them so I can't confirm. Worst case, you can try and easily return to Amazon if they don't fit.
Yes, you can use a 12V battery instead of the 12V power supply if desired.