I acquired a couple of months ago a 8 months old 2nd hand bicycle, an Ammaco Madeira.

It was just 50 pounds, and even if it was not in the best of conditions, I thought that just by changing the brake pads, inner tubes and one wheel cover would be OK.

It was indeed OK for a while after making those fixes, but now the bike is getting worse each time I use it. I try to commute to work with it but it's getting harder and more annoying every day (4 miles to and out of central London).

First, the sprocket -pedals block (forget my lack of proper terminology, I'm a noob when it comes to bike mechanics, but trying to learn!) is a bit loose, and getting looser every day. This is very annoying (feels unstable) when pedalling, makes the chain skip sometimes and in fact snapped my chain recently, which I had to replace. This is a video to make it clear:

https://yfrog.us/28dsc00085z

Is there a way I can fix this by myself? I have basic tools only, but I could buy some new ones if they are not too expensive and completely necessary.

Second, the handlebar is also getting loose at the T-joint. This is due to the metal itself getting dented with the use, as the screws are already tightened to their maximum. I suppose that adding some kind of rubber layer between the metal pieces would fix it, but I don't know if I can buy a special cut exactly for this or should I do it by myself MacGyver style. Again, a little illustrative video:

https://yfrog.us/5mdsc00087z


I would take it to a mechanic if really necessary, but given that:
1) The two bike shops around my area are closed in the morning when I go to work, and closed again by the time I'm back home, and closed during weekends.
2) I work in central London so the mechanics in that area are very expensive.
3) This bike costs new around 180 pounds, and with the fixes I have done to it the 50 pounds that I paid initally are quickly getting close to those 180 pounds. It makes me feel a bit like a moron so I want to fix it for as little as possible.

I would prefer to do this by myself. It's educational too

Thank you very much for any help!

You appear to have a loose or damaged bottom bracket, and it will take special tools but not enough information to tell you which ones. Likely better off taking it to a bike shop. Your stem appears to need tightening with the allen bolt on the side of the stem where it pivots; a hex key shouldn't be very expensive. Hopefully your using the bike with these components loose hasn't caused permanent damage....

On your drive side crank arm (the loose one) there should be a bolt or nut that attaches to the spindle, its probably coveredby a dust cap which you need to either unscrew or pop out (depending on style) you should be able to tighten the nut or bolt with a 14mm socket wrench or an 8mm hex wrench, riding with this kind of slop in the interface can cause irreperable damage to the arm, tighten it down reaal tight and cross your fingers
sorry I can't help you with the stem, I've never worked with adjustables before


*EDIT*
Just rewatched your video, seems the slop is on both sides indicating a loose BB, take it to a shop, you will require special tools

Thank you for your quick input... that sounds bad . Would something like this be expensive to repair in a bike shop, in term of work + possible new parts?

The side bolts are already tightened to the maximum I could with my gloved hands... it's also moving sideways, not just up and down, although to a less extent so it can't be really seen in the video, but you can feel it when riding.

No a bottom bracket adjustment should not be much assming you have a cup and cone bb and its not trashed, if its a cartridge BB youll just need a new one (~25$, sorry don't know what that is in pounds)

Can't really tell by the pic, but there's a good chance that you have a cup & cone bottom bracket, which are quite tinker friendly. sheldonbrown.com, parktool.com and probably bicycletutor.com will tell you all about them. To do a proper job, which would include pulling the crank arms in order to clean, inspect and relube the bearings you need a crank puller. Not that expensive as special tools go, but available in several variations perfect for trapping the unwary buyer.
A "limp-along" bodge job can usually be done with a large plier, a drift punch and a hammer. The drift punch can successfully be replaced with a flat-bladed screwdriver if you don't mind mangling it a bit.

Is this where the bar attaches to the stem?(the bar being the part perdendicular to your direction of travel, and the stem being the part pointing forward.)

If by that you mean that the bar is getting crimped by the clamp I recommend an immediate replacement of the bar. Those "dents" will act as stress risers and can sooner than you think cause the bar to snap off. Maybe not quite as critical on a bike with a fairly upright riding position as on a bike with drop bars, but certainly not a type of injury that I'd like to flirt with.

Don't even try it. Rubber, tape etc is fine for mounting cyclocomputers, smallish lamps and things like that, but it will simply squish for the pressures needed to hold a handlebar in place. If you must, try cutting pieces out if a soda can instead. It's called shimming, and soda cans will work OK for that.

Also, a replacement quill stem isn't expensive. Last one I bought was £12, and for the type you've got I've only ever seen two sizes to choose from. Assuming you can get the diameter of the vertical part figured out you're all set to go shop for a replacement.