Best way to wire 50amp in breaker box
Coming in the bottom is fine.
You can use either neutral bar, you can see a black insulated bar going behind the main breaker that bonds the two. There is no actual restriction saying which side to use, or actually to use the same side as you mount the breaker. The right side of the panel is so congested that to me there seems no good reason to use the bar on the right.
Generally you can mount the size and type of breaker specified on the panel cover any place not prohibited by the instructions on the panel cover. In this panel there are likely no restrictions for the available spaces.
Do as @Harper instructed and make use of the bottom space to loop extra wire long enough to reach anywhere in the panel.
The length of the stripped conductor is not conclusive evidence that the wire is or isn't fully inserted into the lug, but without looking directly at I can't say you shouldn't be concerned. Unless you have what is sometimes called a Fireman's Disconnect Don't attempt to service this yourself. Call a local electrician. He will know the procedure to work with the utility to temporarily disconnect power.
Also note the 2020 NEC requires 240v receptacles to be GFCI protected in locations that only 120v receptacles previously required protection.

210.8(A) Dwelling Units. All 125-volt through 250-volt receptacles installed in the locations specified in 210.8(A)(1) through (A)⁠(11)
and supplied by single-phase branch circuits rated 150 volts or less
to ground shall have ground-fault circuit-interrupter protection for
personnel.