"When a company (especially in Canada/America) ask you to give them your expected salary, do they normally give a counter? I was thinking to give a higher than normal salary expectation ...".
It's different with each employer as to whether or not they will "counter offer" and also upon your definition.
If you simply ask for a dollar an hour more and the employer simply offers a dollar an hour less then it's not so much a situation of counteroffering but one where you are paid exactly what you wanted.
A true counter offer is not a systematic beat down but a negotiation of work vs. pay.
In most cases an employer has a specific job with specific duties unless the position or company are new. This is usually accompanied by a fixed range of pay.
Some places simply offer a bit less, some accept your offer, and few offer more. Seldom will some of the work needed to be done be reduced, occasionally the interviewer will add to the list of tasks to be done beyond what was specified in an advertisement or job description; it's worse when they bait and switch after you are hired.
If they find through interviewing that seemingly suitable candidates either will not do some tasks as part of that job or ask for much higher wages than the intended range then rather trying to offer out of range (tasks or pay) they will close the position and readvertise, sometimes specifying the rate of pay.
There's little productive value in trying to tack on extra duties beyond what such a position would normally call for and realistically nothing is saved by attempting to beat someone down and then hire them - either leads to difficulties or unwillingness to perform as though a good wage had simply been offered up front.
It depends also upon the labor market as to whom holds the reins. With a list of simple tasks that the employer feels anyone could do the likely have a fixed list and rate of pay; you would accept that (and that they're the boss) or they simply go to the next candidate on the list (often thumbing their enormous stack of resumes at you) - don't be fooled into thinking that everyone who applied is a realistic candidate or of equal capabilities. If the pay is low that is a tactic to keep it low.
If it's a difficult job, not a lot of candidates, or pays a fair bit you'll more likely be expected to do all applicable duties normally associated with such a position and similarly it's the employees who will be the ones to set the range; being beat down is as likely to be well received as it's likely to be offered more.
So it depends upon the job, company, and labor market, even the country or culture as to how much negotiations are appreciated. In North America usually they are not but some types of work may well be suited to negotiating duties and pay so in those cases it's expected.
"I am planning to leave for greener pastures, but wanting to see how to handle salary negotiations (in North America, but likely applicable elsewhere).".
There are many types of jobs but fewer types of pay (usually the pay is the same for the same job title; but the tasks performed, or the speed/quality, differ).
There is a cost of living, an amount you want to save, and a premium for difficulty or unpleasantness that set a minimum wage rate for everyone; usually the Law sets a minimum wage that is beneath that level - thus that type of work is probably work you should avoid.
You need to understand how much you are worth to the company if it is a position that brings in the money; some jobs earn the company little or nothing, it simply facilitates a smoother interface between one part and another or saves the owner from doing the work themselves. Again these might be jobs to avoid but there is a realistic minimum that they must pay.
Your situation is another consideration, in your (first) case you wanted your Visa sponsored and now you are beyond that. When building up a resume or a dwindling bank account sometimes you must not ask for top dollar otherwise you should - you rarely hear of places that pay too much.
By knowing your value you are better able to explain your value. If anyone can do the task equally there will probably be too many applications to be worth your while. I have all too often heard that there were 100's of resumes sent and that they'd have to look through them all; I'm making nothing waiting months for that to happen so count me out.
In my case I know that I can do over 10K of work per day - but that requires that they have that much work and don't offer cut rates, instead they need to charge more to slightly reduce the workload and make profit per square foot greater.
If the company hires me, offers good money and the work area is properly setup to do the work then I take the job and money comes in for the both of us.
If the company can afford to mess around then I can not. If it's not too busy I can still pay my own way so they might as well hire since if they don't they'll be losing 1000's per day - if they can afford that they can add it to my paycheck instead of being wasteful.
Many companies have a price in mind that they want to pay and it's like it's a matter of principle that they simply won't pay more no matter what - I've all too often seen the same position advertised and readvertised for months, even years. Obviously there must be some problem there.
When my resume was shorter and I was younger people would call me in not only to ask if I could do work that was the same as outlined on my resume but also if I'd like to work for less than minimum wage.
Now I'm older if the company calls in people categorically similar to me they soon find out that people expect a certain amount (enough to raise a family) and usually cancel the offer of the position - then they readvertise a stripped down set of requirements and specify a low wage (often asking for people with less than 6 months experience). That's certainly not what the first advertisement made the position seem like.
There are very few overpayers, some fair payers and many underpayers. If the company isn't efficient or doesn't price correctly they simply won't have money to hire people who can bring in the customers (instead of repelling them) and bring in the money to keep everything going.
All too often an amount will be agreed upon and work will start but soon after additional unpaid duties will be piled on, so it doesn't matter what they pay if they're going to double the workload and responsibilities with little or no increase in pay.
Bottom line is that if you can't find out what they pay it's a gamble at least find out what the similar place 'down the street from them' pays or a dissimilar place directly across the street - then you can say that if they don't like your offer that you could go work here or there instead.
Letting great people go to the competition is a death sentence for the company, not only does word get out but the competition grows stronger.
Look at it this way: If you don't work there for 40-50 years then they've certainly lost more money than you would have made in a lifetime of working there - so they've paid a lifetime's worth of wages (profit) for you to work elsewhere. If the rebuttal is that they were only going to hire a single person or that they had so many to choose from that your contribution would have been insignificant then that sets the stage for growth potential and employment security.
Look at Sears Canada as an example (yes, apparently they are still hiring despite closing). Not a simple hiring process (see comments at GlassDoor), no feedback, slow advancement (none nowadays) and not a big paycheck - worse is that Benefits, Pension Funds, Bonuses, etc. for people (except upper management) who've worked there decades are not being paid (secured creditors are being paid first).
So cash in hand at a modern and progressive company with growth beats begging for the interview, the job, the pay, the raise, the recognition, the time off, new equipment or repairs.
If they can afford to mess around they're in a better position than you or they're wrong (usually the later, in a company that wants investors (including you) and a prosperous future). Don't work somewhere that seems to thrive on a loss as they may well be a write off for a larger company (or investor) and it's not the mindset of a winner.
The more you have to do, before being hired, during interview(s) and on the job, the more you should get; or cut your loses early and seek a more profitable and satisfying use of your time.
The Internet, Networking, even asking at the backdoor are all ways to find out about a business and what they pay (cash and satisfaction). Know where the grass is greener and mow your way there.
To find out a range of wages try asking at the place in question if it's somewhere where it's possible to access the working employees without being blocked by a gatekeeper. Searching online at places like GlassDoor.com for the place in question or similar businesses in the area is also an effective means to come up with a good estimate.
Exactly how much depends upon factors too numerous to mention but you might want to consider travel time to and from work and the company's 'vision for the future' along with things like benefits and whom you will work with.
Knowing exactly what is required of you is fairly well defined in the US and Canada for many occupations.
In the USA the Bureau of Labor Statistics has the Standard Occupation Classification, and in Canada the Department of Employment and Social Development Canada uses the National Occupational Classification.
If the job description calls for more than what is specified in either of those lists it's reasonable to ask for premium wages as you're doing more than one job (and thus would have had to return to school to learn the additional skills, or acquired training that widely varys in quality and comprehensiveness at another employer).