Two questions on beginner PCB design:
For integrated chips on PCB design, can you draw routes from the side of the contact pad, or is it standard practice to go head-on straight to the pad entry?
For large capacitors that have an output to GND, on a board that has a GND plane, can you put a via directly to the GND plane? Does this help dissipate heat?
You can route your signals to whatever point of the pad that is convenient for you, it does not matter (at least not until you go into very high frequencies). So both options on your image are equally valid (you can even connect with a diagonal line, or on the "inside" side).
What you are drawing on your second image is called a "via in pad": it is perfectly allowed, but requires the via to be filled (epoxy or metal) before soldering the capacitor (otherwise the solder paste will flow into the via, with a great risk of getting a bad solder joint). Most PCB manufacturers can do via in pad. But usually, they charge extra for it (so if you can avoid using via in pad for your design, it's cheaper).
For thermal dissipation, all depends from where to where you want to dissipate heat (from the capacitor itself to the ground plane?). Often, the best solution if you can, is to add a ground "plane" as big as you can on the top layer. If you need additional transfer to ground plane on an internal/bottom ground plane, then add as many vias as you can (if heat comes from the capacitor, one via beneath the capacitor is probably slightly better than one next to it, but vias all around would be even better is you have enough space.
For (1), yes, you can certainly do that. The only case where it might be a problem is when it's a very high-bandwidth controlled-impedance signal, but if it's a high-bandwidth controlled-impedance signal it's not going to be on the edge of a chip anyway.
For (2), this can cause manufacturability issues. I recommend you look up the term "via-in-pad" for more information; in short, the via can wick away solder during reflow, leading to a poor solder joint. There are ways to get around this when you need a via in a pad, but they add a lot to the cost of the board.
Also, if you are having thermal issues with a capacitor to the point that you need to consider heatsinking it, you're almost certainly doing something very wrong.
Yes, though I would avoid going directly between two adjacent pads if they are very close. This looks like a short-circuit when soldered, making it harder to spot any real short-circuits.
If the pad is large compared to the hole, via in pad area is fine even without filled vias. My own rule-of-thumb is that filling is needed if hole diameter is more than 10 % of pad size. For example, a 3x3 mm pad with 0.3 mm via is fine, the amount of solder that may wick into the hole is insignificant.
