Bash – Run entire bash script as root or use sudo on the commands that need it

I think you missed a level of quotes escaping. At this high level of escaping, it is best to simply make a little script for every stage where otherwise quotes would be required.

Otherwise, you could try this modified version (but mind you, I don't encourage this coding style!)

function _remoteInstallation(){
    local retval=1
    local debToInstall=$(basename "$1")
    local remoteMachine="$2"
    spawned=$(expect -d -c "
          set timeout 1800
          spawn \"/usr/bin/ssh -t borrajax@$remoteMachine /usr/bin/sudo /usr/bin/dpkg -i /home/borrajax/Documents/$debToInstall\"
          expect {
                \"Are you sure you want to continue connecting\" { send \"yes\r\"; exp_continue }
                \"password\" { send \"myPassword\r\";  exp_continue }
                \"[sudo] password\" { send \"myPassword\r\";  exp_continue }
                default { exit 1 }
          }
    " )
    retval=$?
    return $retval
}

Q#1: Will I only be prompted for a sudo password once, or will I need to enter the sudo password on each invocation of a command inside the script, that needs sudo permission?

Yes, once, for the duration of the running of your script.

NOTE: When you provide credentials to sudo, the authentication is typically good for 5 minutes within the shell where you typed the password. Additionally any child processes that get executed from this shell, or any script that runs in the shell (your case) will also run at the elevated level.

Q#2: is there still a possibility that the sudo permissions will time out (if, for instance, a particular command takes long enough to exceed the sudo timeout)? Or will the initial sudo password entrance last for the complete duration of whole script?

No they will not timeout within the script. Only if you interactively were typing them within the shell where the credentials were provided. Every time sudo is executed within this shell, the timeout is reset. But in your case they credentials will remain so long as the script is executing and running commands from within it.

This limit is policy-specific; the default password prompt timeout for the sudoers security policy is 5 minutes.