The 700-406 model looks to be a standard ATX PSU and mATX motherboard, so I'd suggest getting the best graphics card possible within the budget.
A PSU swap is definitely needed for any half-decent graphics card, and the extra cooling required for a more powerful graphics card needs to be considered.
Transferring everything into a new case with better airflow is one way to do it and that would work well, but it would eat into your budget.
Another option is to buy a graphics card that's designed to vent most of its hot exhaust air directly out of the expansion slot, putting much less strain on the internal cooling than a regular card would.
Here's an 8GB RX580 and 6GB GTX1060 that would do just that:https://www.newegg.com/Product/Product. ... -_-Producthttps://www.newegg.com/Product/Product. ... -_-Product
In some games, a 1060 6GB is faster, in others an 8GB RX580 is faster. Overall the RX580 is faster, cheaper, and uses more power, but they're both pretty comparable.
Pair that with something like this Seasonic 80+ Gold Modular PSU on offer for the next six days:https://www.newegg.com/Product/Product. ... -_-Product
As for your other questions:
1) It is safe to transfer all the old components to a new case?Yep, it looks like everything is a standard component that will transfer without any unforseen problems. When you buy a case most of the screws and fittings you will need to do the transfer come with the new case, the rest you will be able to salvage from the old one.
2) how do I pick out a case?Choose a size, choose a budget, pick the one you like the look of. There are very few bad cases as long as you stick to a reputable brand. There's plenty of high-quality choice in the $50-100 range, and if you tell people here what sort of case you like I'm sure we can make some suggestions.
3) what should I know about transferring parts to a new case, and what are the riskWatch a youtube video on how to build a PC. That'll cover everything you need and more. Techreport has its own video somewhere, I think it's linked on every one of the system guides that come out a couple of times a year.
4) what are things I need to know about this whole project so nothing breaksIt's quite hard to break a PC if you're careful, but my advice as a beginner is to treat everything as if it were brittle glass that is static-sensitive so don't force anything if it doesn't fit the connector, if you're sure it's the right plug into the right socket and the connection is causing the board or card to bend, provide some support from behind the board so that you're not stressing it (this mainly applies to power connectors on the graphics card and motherboard, as they can be stiff. When you're unsure, Google it rather than guessing
Also which 1050/1060 should I buy? There are many iterations and I don’t know which one is which?I've linked a couple of cards that would work without you needing to change to a new case. If you're really keen on getting a new case, try to get the best graphics card you can afford with your remaining budget. That means you want a 1060 instead of a 1050, and you want the 6GB version of it instead of the 3GB version. More expensive does not necessarily mean better, either. A lot of the more expensive models are more expensive because of RGB Lighting and very minor factory-approved overclocks, but in reality what affects performance is mainly how big the card is, because a larger card usually has more surface area for better cooling.