First though, as for hiring a full time employee with only the ability to pay them for two months: if you're not up-front with this person you're a real jerk. I've solved this by bringing people in for contracts that are the duration and explaining that future contracts or conversion are dependent on more work coming in. Yeah, it's effectively the same except for in my scenario you may be offering the job to someone with dependents who is risk averse and definitely needs that paycheck in month three. This basically gives them the risk signal they deserve without you having to say "we've only got this much money and no idea what's going to happen in month three." I'll do almost anything before I screw up someone else's livelihood.
As for the contracts, I've gotten good enough to read most of them competently. I redline the sketchy parts and the parts my lawyers have redlined in the past. Submit my alternatives or reason why it should just be struck and keep going. It's generally around things like 100% payment at the end changed to milestones or 50/50, liable for all damages changing to liable for up to the value of the contract, etc.
There are some things you really can't mess with. If you've got a megacorp client and their AP policy is 90 days, don't bother. You're not getting it for AT LEAST 90. Likely 5-6 months if you're persistent/have a good project owner on their side. They will never agree to a change, and you're not going to sue them for a late invoice unless we're talking about large sums of money that would've dictated this is all being handled by lawyers anyway.
Other things get really tricky. Some will slide in IP ownership/transfer clauses that could essentially mean they have claim to anything your company's ever done. I've got some well written reaponse clauses that I re-submit with- but if they push back on those at all I instantly walk away.
So, yeah, do read the contracts, focus on a few key areas and look for over reach or hostile clauses and submit sane, fair responses that can be easily accepted. It's fine if you have a lawyer doing all this for you in an advisory role, by be the person who fronts the requests and make yourself part of the process. You'll learn the patterns very quickly and will need to pay your lawyer less frequently as well as you'll learn the counter-proposals so you can shape your proposals to reduce the counter proposals in the first place.
Let the esoteric stuff work itself out in a lawsuit, stuff has probably really hit the fan in that case.
As far as damages for outages and stuff, you have to assume that if you're working on a smallish (sub 500K) there's absolutely no way they're suing you for a few hours of down time if you provide good customer service during that period. I've had HUGE F ups that led to 6+ hour outages and still received highly positive post project feedback from clients because we had such a good relationship before our mistake(s), we were highly communicative and forthright during the issue, and provided a good post mortem quickly after resolution.
To that end, if you find that you often have like-warm relationships with clients, maybe you shouldn't take the risks and just have a good lawyer handy.