Bollox to you, that's a perfect answer to those questions :)
Mine woulda been funnier tho. :D
If there's no "producer" around, and the bandleader's just a singing guitarist who wants his vocal and the guitar up loud (all over NOOL, anyone who's met this guy is nodding and growling inwardly some mantra about hanging for musicians), the engineer ends up being the producer. Sometimes good, sometimes bad, it depends on one key factor.
If the Engineer is a musician, he will produce well, and be sympathetic to the music. If he's not, he' pish you about from pillar to post to get stuff "right" and/or convenient for his lazy ass.
Live engineers are something between a producer and a concert conductor.
While there are many who bash some faders up and get a nice noise, then see if there's any collateral minge to be scored among the groupies. There are plenty who are RIGHT in there, hands on, flying the console along with the music, sometimes awesome, like New Order in the 80's, and sometimes fucking terrifying, like New Order in the 80's. basically depends on whether the discksack knows his band's material, and how much he's had to drink.
Not all bands tour a full sound crew. Very often they will take only a Front of House engineer (the guy you see) and/or a monitor engineer [1] (the poor bugger, almost always stage right, who mixes the various different wedge monitor and sidefill feeds for the band, in small venues, the FoH engineer will do that from their desk, but as rig and venue size increase, and communication gets harder, the need for a mug, I mean monitor engineer increases too). Sometimes they take neither.
If you;re going to be world hopping, and you don't need your desk man to "play" with you, it's better to just use guys who come with the rig you'll hire in each country. Likewise at festivals, some bands will turn themselves over to the festival head engineer, most will have their own guy there to have fights with him over the console, it's not really a 2 man job, but they struggle through with one knowing the rig and one knowing the band.
If you NEED your man, he has to go with you, and if you're playing in different sorts of venue, (say you were in pubs last week, and yer doing Wembley this week), he'll have to wrk with someone from the hire company, at least for a couple of shows til he gets the picture.
if you are U2
You block rent half of britania row, or whoever's inventory for 1 year and pack it into 747's with your entire sound and light crew. (such lower roadfilth as can't be found in the homesless hostels and drug clinics at the destination, are sent 3 to a teachests). they probably also take their own gorillas, foot rubbers, hairdressers etc.
real world, it's you and your mate who owns a mixer in a van that stinks of piss going somewhere to plug into the in house disco rig and hope the glitterball and moonbeam work.
I know, i've been that mate :)
[1]if tony Robinson ever gets that "worst jobs in history" up to the modern music industry, monitor engineer's a major candidate. it's a cruel and thankless task, and only worth undertaking if you have the temper of Buddah afer he's toked an ounce of temple ball, and skin like a chav Rhino that's been on the sunbeds. Never give the job of monitor engineer to a psychotic or someone who carries weapons, unless the band commonly carry weapons on stage, in which case put his console in a tank or only give the job to "expendables".