Yes you can connect several Li-Ion cells in parallel but before you do so, check that they have (almost) the same voltage.
If you buy several at the same time from the same supplier, changes are this will be the case.
If the voltages are more than 0.2 V (I just sucked this value out of my thumb !) different, you have to balance the cells. Either charge them fully with the same charge, after charging they will have the same voltage. OR you connect them in parallel but with a small value resistor between both + poles. A 100 ohm resistor will do. This resistor will limit the current flowing from one cell to the other while they balance themselves. When there's 0 V across the resistor left, the cells are balanced and you can connect the + poles also without the resistor.
If the 2.5W solar panel will be enough, depends on your patience ! Using 2 cells will double the charging time. Using a solar panel with double the power will halve that charging time again. But the charging circuit you're using can only supply up to 1 A so it makes no sense to use more than 2 2.5W, 5V (so 0.5A) solar panels.
Panasonic are excellent batteries ! Also Samsung and LG make excellent batteries. I would not recommend most cheap UltraFire. Either get cells with "solder tabs" to connect them in parallel and to connect wires to them OR get cells without "solder tabs" but then get a battery holder. You should avoid to solder directly on the battery.
I would recommend getting "protected cells" (these have a small battery protection circuit) without "solder tabs" (most protected cells do not have these anyway). And to use a battery holder, if you ever need to replace the batteries, it will be easy.