Can I replace a Shimano RD-TY15 with Shimano Tourney TX?

Because it's been out of production for some time. There's no useful info to be had anyway. It is adjusted the same as most any rear friction derailleur. It is not useable with indexed levers (click to shift). If you really think you need info do the following Google search: "RD-TY10 site:shimano.com" and scroll through the results.

I’m happy to help further over the phone at https://www.6ya.com/expert/donald_f2ed37026a3ac881

This may be to do with the replacement having a shorter cage, so it can't go on the big sprocket and big chain ring at the same time.

Try to not use the big rear sproket and the the big front chaining at at the same time,

If you can't find with a search for st-ef40 diagram or on the Shimano site then it does not exist. Although not identical carefully disassembling the other shifter may give you enough insight.

I’m happy to help further over the phone at https://www.6ya.com/expert/donald_f2ed37026a3ac881

Shimano Tourney parts are one of their lower end component groups and as such are not really designed to be serviceable. Shimano uses a system of "controlled obsolescence" so that their components cannot be repaired. they are inexpensive enough to just replace them, and with all of the plastic that they use these days it's not worth the effort when a new derailleur can be had for $30.

I'm confused, You show a rear derailleur and ask a question about a front one? Cable for front usually goes from bottom up, through an eyelet then into the holding bolt. Make sure your cable is in the low gear setting and your chain is on the smallest gear. Pull cable tight, and secure. If you have problems, take your bike to the nearest bike shop, they can help you

Snagging? Like hooking on the teeth in a bad way? Go through all the rear derailleur adjustments.

http://www.parktool.com/blog/repair-help/categories/derailleur-systems

Look for worn parts...

http://sheldonbrown.com/chains.html

Get out your wallet.

Simple web searches will answer many questions like yours immediately.

http://www.parktool.com/blog/repair-help/front-derailleur-adjustments

Look around here.

http://techdocs.shimano.com/techdocs/blevel.jsp

Mountain Shimano Tourney Shifters

This problem is best taken into a bicycle shop to have them look at it. Could be as simple as adjusting the cable tension, or as complicated as parts not in the correct location or functioning properly. The shop will be able to make a quick diagnosis once they can actually put hands to it.

Go to shimano.com and download instructions.