Concise answers to your questions:
Depends on the loan and the bank; when you "accelerate" repayment of a loan by applying a pre-payment balance to the principal, your monthly payment may be reduced. However, standard practice for most loan types is that the repayment schedule will be accelerated; you'll pay no less each month, but you'll pay it off sooner.
I can neither confirm nor deny that an internship counts as job experience in the field for the purpose of mortgage lending. It sounds logical, especially if it were a paid internship (in which case you'd just call it a "job"), but I can't be sure as I don't know of anyone who got a mortgage without accruing the necessary job experience post-graduation.
A loan officer will be happy to talk to you and answer specific questions, but if you go in today, with no credit history (the student loan probably hasn't even entered repayment) and a lot of unknowns (an offer can be rescinded, for instance), you are virtually certain to be denied a mortgage. The bank is going to want evidence that you will make good on the debt you have over time. One $10,000 payment on the loan, though significant, is just one payment as far as your credit history (and credit score) is concerned.
Now, a few more reality checks:
$70k/yr is not what you'll be bringing home. As a single person without dependents, you'll be taxed at the highest possible withholdings rate. Your effective tax rate on $70k, depending on the state in which you live, can be as high as 30% (including all payroll/SS taxes, for a 1099 earner and/or an employee in a state with an income tax), so you're actually only bringing home 42k/yr, or about $1,600/paycheck if you're paid biweekly. To that, add a decent chunk for your group healthcare plan (which, as of 2014, you will be required to buy, or else pay another $2500 - effectively another 3% of gross earnings - in taxes). And even now with your first job, you should be at least trying to save up a decent chunk o' change in a 401k or IRA as a retirement nest egg.
That student loan, beginning about 6 months after you leave school, will cost you about $555/mo in monthly payments for the next 10 years (if it's all Stafford loans with a 50/50 split between sub/unsub; that could be as much as $600/mo for all-unsub Stafford, or $700 or more for private loans). If you were going to pay all that back in two years, you're looking at paying a ballpark of $2500/mo leaving just $700 to pay all your bills and expenses each month. With a 3-year payoff plan, you're turning around one of your two paychecks every month to the student loan servicer, which for a bachelor is doable but still rather tight.
Your mortgage payment isn't the only payment you will make on your house. If you get an FHA loan with 3.5% down, the lender will demand PMI. The city/county will likely levy a property tax on the assessed value of land and building. The lender may require that you purchase home insurance with minimum acceptable coverage limits and deductibles. All of these will be paid into escrow accounts, managed by your lending bank, from a single check you send them monthly. I pay all of these, in a state (Texas) that gets its primary income from sales and property tax instead of income, and my monthly payment isn't quite double the simple P&I.
Once you have the house, you'll want to fill the house. Nice bed: probably $1500 between mattress and frame for a nice big queen you can stretch out on (and have lady friends over). Nice couch: $1000. TV: call it $500. That's probably the bare minimum you'll want to buy to replace what you lived through college with (you'll have somewhere to eat and sleep other than the floor of your new home), and we're already talking almost a month's salary, or payments of up to 10% of your monthly take-home pay over a year on a couple of store credit cards. Plates, cookware, etc just keeps bumping this up. Yes, they're (theoretically) all one-time costs, but they're things you need, and things you may not have if you've been living in dorms and eating in dining halls all through college.
The house you buy now is likely to be a "starter", maybe 3bed/2bath and 1600 sqft at the upper end (they sell em as small as 2bd/1bt 1100sqft). It will support a spouse and 2 kids, but by that point you'll be bursting at the seams.
What happens if your future spouse had the same idea of buying a house early while rates were low? The cost of buying a house may be as little as 3.5% down and a few hundred more in advance escrow and a couple other fees the seller can't pay for you. The cost of selling the same house is likely to include all the costs you made the seller pay when you bought it, because you'll be selling to someone in the same position you're in now. I didn't know it at the time I bought my house, but I paid about $5,000 to get into it (3.5% down and 6 months' escrow up front), while the sellers paid over $10,000 to get out (the owner got married to another homeowner, and they ended up selling both houses to move out of town; I don't even know what kind of bath they took on the house we weren't involved with).
I graduated in 2005. I didn't buy my first house until I was married and pretty much well-settled, in 2011 (and yes, we were looking because mortgage rates were at rock bottom). We really lucked out in terms of a home that, if we want to or have to, we can live in for the rest of our lives (only 1700sqft, but it's officially a 4/2 with a spare room, and a downstairs master suite and nursery/office, so when we're old and decrepit we can pretty much live downstairs). I would seriously recommend that you do the same, even if by doing so you miss out on the absolute best interest rates.
Last example: let's say, hypothetically, that you bite at current interest rates, and lock in a rate just above prime at 4%, 3.5% down, seller pays closing, but then in two years you get married, change jobs and have to move. Let's further suppose an alternate reality in which, after two years of living in an apartment, all the same life changes happen and you are now shopping for your first house having been pre-approved at 5%.
That one percentage point savings by buying now, on a house in the $200k range, is worth about $120/mo or about $1440/yr off of your P&I payment ($921.42 on a $200,000 home with a 30-year term). Not chump change (over 30 years if you had been that lucky, it's $43000), but it's less than 5% of your take-home pay (month-to-month or annually). However, when you move in two years, the buyer's probably going to want the same deal you got - seller pays closing - because that's the market level you bought in to (low-priced starters for first-time homebuyers). That's a 3% commission for both agents, 1% origination, 0.5%-1% guarantor, and various fixed fees (title etc). Assuming the value of the house hasn't changed, let's call total selling costs 8% of the house value of $200k (which is probably low); that's $16,000 in seller's costs. Again, assuming home value didn't change and that you got an FHA loan requiring only 3.5% down, your down payment ($7k) plus principal paid (about another $7k; 6936.27 to be exact) only covers $14k of those costs. You're now in the hole $2,000, and you still have to come up with your next home's down payment.
With all other things being equal, in order to get back to where you were in net worth terms before you bought the house (meaning $7,000 cash in the bank after selling it), you would need to stay in the house for 4 and a half years to accumulate the $16,000 in equity through principal payments. That leaves you with your original $7,000 down payment returned to you in cash, and you're even in accounting terms (which means in finance terms you're behind; that $7,000 invested at 3% historical average rate of inflation would have earned you about $800 in those four years, meaning you need to stick around about 5.5 years before you "break even" in TVM terms).
For this reason, I would say that you should be very cautious when buying your first home; it may very well be the last one you'll ever buy. Whether that's because you made good choices or bad is up to you.
