First, I'm not sure about whether they email you the boarding pass because we simply decide when we're ready to print and go ahead and do it (Ryanair won't let you before 15 days pre-flight but I don't know ref EZY) ..... but you can either print directly from the screen when you open it from the relevant part of your 'manage my bookings' OR save as a pdf for printing later.
Ref the scramble.... there's still the scramble, but if you have a seat booked the only thing you need to consider is space for your hand luggage in the overhead lockers. So you still might want to be early in the queue for that.

If they board passengers by row numbers, as a lot of airlines do, then it doesn't really help to be early in the queue - if you are in the final batch of seat rows to board, a lot of the space in the overhead lockers tends to have already been filled up by other passengers.

I've never been boarded by row numbers by EZY or RYR, it's just (Priority Boarding/disabled etc notwithstanding) 1st queue first served.

Easyjet don't board by seat rows as Fiona says it is Speedy Boarding, those requiring assistance or with young children then the rest. I don't myself think the rush is as bad as it used to be as everyone has an allocated seat but of course there is still a bit of a surge. They seem to be quite keen on cramming in the baggage into overhead lockers however my case fits the old standard size rather than the new smaller one that is guaranteed to go in overhead locker, however so far not had any problems getting mine in or had to put it in the hold. Easyjet you can print of boarding card 30 days prior to flight, very simple once you have entered your id details and if lost prior to flight you can go back into site and re print.

I know that's how they always did it before everyone got a seat allocated, but I wondered if they might change it now. I normally fly with Monarch and they always board passengers in batches according to seat rows.

When we flew Faro/Luton last week easyjet did it in 3 batches, left hand queue priority boarders + invalids + infants to be followed by rows 1-16. Right hand queue rows 17-30. We were allocated 29E&F and knowing the hostesses stuff their bags in that area we stepped smartly to squeeze our two undersized cases in the overhead above our seats. The Johnny come lately's around us had their cases taken from them with the threat of delays to put them in the hold. However by removing coats, handbags and duty frees to underneath the seats they managed to squeeze them all in the cabin. Poor chap in 29D had his case above row 15 and his wife in 30D some other row down the front. You can imagine the chaos on landing when he tried to move forwards to recover their luggage. Of course this opened up our exit and we were 2&3 down the back stair. Despite being parked as far from passport control as is possible to be we were clear of arrivals and at our pick up point in 15 mins that is the beauty of regional airports. Of course it does help to have e passports!

Not sure about other airlines but easyjet have now come out with an app for smart phones or tablets whereby you can download direct to phone or tablet your boarding pass which can be used no need to print off. Which may be useful given availability of free wifi most areas.

No because everyone wants to make sure they get the locker space above their seat. If you sit near the back those spaces are often already filled with air hostess stuff.
Regarding the size of cabin luggage, my husband's was a fraction bigger because of the wheels and Mrs Jobsworth said she would let him through This time!
Is the same with all budget airlines that you can only book in online if you pay to choose a seat?

Not so you can check in on line , if you have not paid for a specific seat you will be allocated one at no cost.

Agree not the case easyjet now allocate a seat but it is not certain that if you are a couple say that you will be sat together, to be certain you can pay for that. However even before this when seats were not allocated you could still book in online.
As for jobsworths if they allowed your husband on against the rules how were they a jobsworth?? they do state the size allowed full dimensions including wheels! someone has to ensure flight rules are applied hardly being a jobsworth just a job for me. I will add I had a daughter that worked in the airline industry long unsociable hours and tolerate abusive people who through no fault of hers failed to comply with the conditions of flying .

Where ever I've been on holiday I have so far managed to access an internet shop of some description and printed off my boarding pass.

As for seating at least we are allocated a random seat in advance, even if we don't pay for one (Ryanair, all genuflect) However I noticed that at Valencia airport a few days ago, 80+% of passengers were queing up for about one hour twenty minutes and sweating buckets whilst tutt-tutting to one another! Surely they cannot all be bussiness high flyers, holding diplomatic boxes or on missions of mercy? So I sat there in comfort and watches them and was one of the last to enter the plane and there was my seat. A few minutes at the other end to collect a case is no big deal.

The Ryanair charm offensive appears to be working. I flew Stansted to Valencia last week Despite being a full flight there was no great rush to get on the plane, the staff were helpful and the baggage restrictions were much less stringent.
I suspect the true test will be the return leg from Valencia.

We bought our hand luggage ate the same time, and they were the right dimensions, and they fitted in the test frame at the airport.
The staff said mine was fine but measured my husband's, and the handle on top made it a fraction higher than mine,She said, in a school marm voice" I 'll let you on this time"
Meanwhile hand luggage which was obviously bigger than regulation passed through unchallenged.
So, yes, I do feel she was a Jobsworth, or perhaps pedantic is a better word.

I agree ,travel with a soft bag that will fit under the seat in front of you if overhead space is unavailabe, have a coffee and watch the majority of your fellow passengers get stressed out.

I just can't see how if your husbands case was over size no matter how marginal and she let it go as hold luggage she was being either a jobsworth or pedantic. Maybe if yours was the only oversize one not allowed on, as you mention allowing clearly oversize ones on as well. Having been on the position myself of having a job which meant confronting the public with what they deemed to be trivial matters I have been called far worse, as I am sure has the school marm person that let your husband take his case on board.