FIRST PART:
As far as moving/leaving/rolling it... you do some research:
1. Is the 401k at a good financial company like Vanguard or Fidelity? This is a strong plus. If it's not with someone like that, then you probably should shift it over to them. There will likely be a transfer or closing fee depending on how big a jerk the former financial institution is. Still worth it if it's a poor 401k plan to get away from them.
2. Do you have to pay any management fees for the privilege of having the said account if you leave it with them? Most don't, but there are some smaller crummy ones that charge you $50 or more per year. This is stupid and totally not worth it mostly (only time would be if #3 applies).
3. Do you have access to some super crazy awesome institutional funds that have great expense ratios due to being with the company's 401k group? Most of the time this is also a no, but it depends on how big and amazing your company is to have brokered a killer deal for the offerings.
4. Do you have a limited range of choices for investments in the 401k plan currently? STRONG reason to do a rollover IRA then, as you'll have access to invest in anything available.
5. Do you have a new job already lined up? If not, wait to do anything until you assess their 401k package. You can sometimes roll an old 401k into your new employer's 401k, and if they have a killer setup available, it would be a great option if they allow it. (once you do a rollover IRA, you can't go back so don't do anything with it until you're sure)
There are a few reasons to keep it as a 401k, so you'll have to figure those out too: better protection in some states against lawsuits/bankruptcy mostly, but there's a few other reasons I can't remember at the moment.
Most likely, you will end up coming out ahead to move the 401k to Vanguard or Fidelity, creating a rollover IRA and setting up the asset allocation that you've figured out for yourself.
SECOND PART:
The match is considered a benefit to working there, much like bonuses. They're meant to be an incentive to come to and then stay with the company. If you leave before the money hits your account, you're likely screwed yourself unless you have specific paperwork showing you are to be paid the match if you are employed by them as of such and such date.
And I asked my contact about when our match would happen with the excuse of "trying to get my yearly investment plans figured out and need to know any details about when it would happen."
My employer paid our match out roughly in February. I read my employee handbook cover to cover, discovered the fine print that said match to be paid to all employees that were working for our company as of December 31 of the previous year. I did not turn in my notice until the money hit my account tho. Better safe than sorry. Same thing for any bonuses too - you gotta be there most likely to get it, so wait till it's in your account (and I'd wait at least a few weeks after receiving it before turning in a notice if there's no firm policy in the handbook).
THIRD PART:
Do all the medical things that make sense if you have killer insurance right now.
Read the employee handbook cover to cover to figure out any benefits, PTO and how it will be used/paid out. You could use the "I'm reading the employee handbook and wanted to clarify something" excuse to ask a few questions without being really suspicious sometimes, but you'd have to gauge your office environment to see if this is a possibility.
If you have contacts you'd like to keep up with, make sure and save the info to a personal file (not valid if we're talking non-compete stuff, just mean folks you have relationships with that you'd like to stay in touch with as a friend stuff), copy over any files or information stored on business laptop/phone etc.. that are examples of work you were proud of or exchanges you want to remember (again, not meaning "steal from your employer" stuff, but like charts or docs you created that help you do your job that aren't proprietary to the company)
Figure out vacation days and take them if you like, or save them up for payout if that is company policy upon separation. If you have a company phone as your main phone, you may want to get your own personal phone ASAP so you can start transitioning over easily (they won't let you keep your phone/number most likely).
Sharpen up your resume, and make sure you have some decent references. ASK a few people if they would be a professional reference for you in the future if you start job hunting. Don't ever assume it's okay to use a person without asking, and be damned sure that they'd give you a glowing reference before asking them. And only people you are 100% sure won't rat you out if you work with them right now.
Do not slack off on your job while looking for another, even in the notice period. Stay classy and do good work right to the last day. Start getting notes on things you do in your job to help with the transition for the new person. Organize your files, and clean up your computer (especially if you store personal files on there too).
It almost never is a good idea to take a counter offer to stay. Once you turn in your notice, if you do accept their counter, you burn your bridge with the company that wanted you, your old company now know you were unhappy enough with something to be looking to leave and you are branded as "disloyal" and likely will be the first one to go if they start looking for someone to boot.