Is it safe to design board at fab houses minimum trace width?

A reputable PCB house will solidly deliver their minimum standard track & spacing.

Some will let you try finer at your risk and some will reject your work outright if under spec.

I assume that you are using a temperature controlled iron. If not, do.

If you are not confident in your own soldering abilities and you have plenty of room then fattening up tracks and especially pads associated with through hole parts does no harm. I'd give the through hole headers as much copper as you reasonably can - especially if the board is NOT PTH (plated through hole). With PTH you get substantially more strength. If the board is single sided (probably not, but ...) then you want to take great care with through hole pad soldering and due care with everything else.

SMD part pads should be suited to the part and you have to learn to accommodate them rather than them accommodating you.

PS: Others feel free to contradict or improve anything I've said. I have very substantial soldering experience but there is always good stuff and ideas to learn. ________________________________

Related: This excellent reference -
TI Analog Engineer’s Pocket Reference - 4th edition
as well as a vast amount of other useful material
it provides some useful information on PCB track current/ voltage drop / heat / fusing issues. Especially pages 55-68.

Do you need to go that small? If so, go for it -- it's advertised, and they will deliver. If not, and especially if the assembler (you?) has little soldering experience, go bigger.

Larger traces are easier to solder to, and are better laminated to the FR-4, so you can, but still shouldn't, overheat areas with less risk of traces lifting. I usually don't go below 10 mil (0.25 mm), as it conveniently matches the finer-pitch leads. Minimum features sizes are useful when necking down to stick a trace between other features (between leads, BGA pads, etc.).

Assembling 20/20 mil boards makes me happy.

If they give 6/6 as their specs, then yes, it's safe to design to that.

However, I wouldn't unless I really need the space. I use 8/8 as my own personal design rules unless there is a good reason not to. That way, any board house in the world can produce the board without problems or extra cost. Note that you can get tighter design rules from a number of places, but as the dimensions get smaller the cost goes up. Just because your board house of the week can do 6/6 before charging more doesn't make that a good design decision unless you need the extra space.

I didn't look up the board house you mention, but $10 for any amount of board is suspiciously cheap. Does that include solder mask, silk screen at least on top, electrical testing? These can save a lot of trouble and human screwups later. Maybe you're OK without these, but you should make sure you know what you're getting. Personally, I wouldn't get boards without solder mask and silk screen on at least one side.