Electrical – Will this 18650 tp4056 parallel circuit work for charging and simultaneous power supply or will things break

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.

This is either dangerous, not advisable or doable depending on...

It comes down to what the "cells" are made up of. Since a bare single cell doesn't have a TH, you are talking about a pack. It may be a pack of 1 single 3.7V cell, in which case it's effectively a cell and you can, to a limit, connect in parallel to your heart's content.

I'd advise you to balance them out to each other with an extra resistance before you hard connect them, though. If they're large cells and 0.5V apart that will mean large currents will flow until they "agree", which is good for neither the low one, nor the high one. It would look a bit like this:

^{simulate this circuit – Schematic created using CircuitLab}

You should pick a TH contact, since it is a temperature sensitive resistance that will be measured, if you connect multiple in parallel it will measure a low resistance, compared to what's expected. Since usually they are NTC 10k, that means the charger will think your batteries are hot when you first connect them, which means it will not charge.

If you have a pack of more than 3.7V, such as 7.2V or 11.1V, then connecting multiple in parallel without any "internal cross connections" will increase the speed at which the first cell will die. If the pack is multiple cells in series without any balancing connections, it can be debated whether that's advisable to start with. But anyway.

If you have a 2 cell in series pack, you will want to connect them like this:

^{simulate this circuit}

This way the odd batteries will join force as will the even batteries and that will severely decrease the statistical chance of a dangerous defect in the pack.

((Of course it's better to start out with resistors for the parallel connections for the first hours to cross-balance again))

If you have a multi cell pack and can't make the cross connections as drawn above, I'd say, on balance, you're better off not connecting anything in parallel at all. Especially for charging purposes.