What I can see from the pictures: many solder joints are not solderd through the board, solder is not showing on the top of the PCB when solderd from the bottom side. This indicates the soldering iron temperatire has been too low or that the solder periods have been too short, so there may be "cold joints". With perfect joint, both top and bottom side will look more or less the same. Now, there may be solderings that look OK from the bottom side, but actually have a bad or no connection. I also see lots of residuals from soldering. It may menas small, thin conductive streaks across where they should not be. I suggest you try brushing / cleaning this off after soldering. Do you know someone with more soldering experience that can help?
Measuring capacitors: you need a capacitance meter. Some DMMs have this. Better DMMs can measure both very large and very small caps. Simler DMMs can only measues intermediate sizes (like 1nF to 100uF) It is best to measure components off the board, as the circuit always influences measurements. If measurements look OK on the board, they may be OK, if they do not look OK, it may be other compoennts influencing the measurements. So it is hard to draw conclusions if they are still mounted in the circuit, unless you have lots of knowledge and experience and can draw conclusions from the schematic. Plastic and ceramic caps are seldom damaged unless badly abused, while electrolyths can be damaged if mounted with reverse polarity or overheated.