Random thoughts:
You don't want to run off the wort from your mash tun and heat that to the next step temp because it would barely budge the overall temp of the mash when added back, and you definitely don't want to boil the wort because you would kill off your enzymes.
You can do that...people do it all the time.
If you infuse hot clean water then it's called an
infusion.
If you pull wort from the mash and heat it and then return it to the main mash then it's called a
decoction. Decoctions are often boiled for a while.
Many people do a mash-out decoction because it would take too much water to do it via infusion and you'd be using up your precious sparge water. So they just pull a thin-mash decoction consisting of mostly or all liquid...heat it and possibly boil it...then return it. EDIT: Thin-mash decoctions work well for mashing-out because the wort is already converted. If you were using a decoction to increase the temp earlier in the mash (before it was all converted)...you would typically pull a thick-mash decoction. Most of the enzymes are in the liquid.
Remember that it takes a lot more hot water to go from 158 F to 168 F than it does to go from 108 F to 118 F...even though they are both a 10 degree increase.
Luckily, when you do lower temperature rests they typically benefit from a thick mash and as you increase the temp the mash benefits from being thinner so infusing your way up works very well...thick at the beginning and thinner at the end.
If you want to practice it...you could try starting at something like 145 F for 30 minutes then infuse (or decoct) up to 155 F for 30 minutes. Even if you miss by a few degrees it should still be fine.
The wort going into the kettle doesn't have to be
that clear, just no grains/husks/chunks. If it's got some really fine particles in it that's OK...you don't want it really cloudy...but it doesn't have to be super clear either. Besides...it will clear more as you go so only the very first runnings will be that turbid (and it will all settle out...the beer will be crystal clear).